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THE ALGEBRA 

OF 

INVARIANTS 



aonUon: C. J. CLAY and SONS, 

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MARIA LANE. 

ffilnsaofa: BO, WELLINGTON STREET. 




Ifipjiji: F. A. BBOCKHAUS. 

i^ffa gork: THE MACMILLAN COMPANY. 

ISombne nnU CTalnitfa: MACMILLAN AND CO., Ltd. 



[All Rights reserved.] 



THE ALGEBEA 

OF 

INVARIANTS 



BY 

J. H. GRACE, M.A. 

FELLOW OF PETEllIIOUSK 

AND 

A. YOUNG, M.A. 

LECTUKER IN MATHEMATICS AT SELWYN COLLEGE, 
LATE SCHOLAK OF CLAKE COLLEGE 



CAMBRIDGE: 

AT THE UNIVERSITY PRESS. 

1908 



Cambrtligc : 

PRINTED BY J. AND C. F. CLAY, 
/^'aK the ONlVE^Slit; PB'BJS. ". 






/ 



Engineerinfj & 
Mathematical 

Terences 

Library 



CO 
CD 

CO 



'StOl 



P PREFACE. 

ri^HE object of this book is to provide an English introduction 

to the symbolical method in the theory of Invariants. 

^ It was started as an attempt to meet the need expressed by 

^;^ Elliott in the preface to The Algebiu of Qualities — 'a whole book 

which shall present to the English reader in his own language 

Nj a worthy exposition of the method of the great German masters 

^ remains a desideratum.' Since then the need has been partly 

^ met by the article 'Algebra' by MacMahon in the Supplement 

v^ to the Encyclopcedia Britannica. The subject has been treated 

from the commencement in order that readers unacquainted 

with Elliott's treatise or any presentation of the elements may 

4s be able to understand the argument. Such readers should bear 

fM^ in mind that this treatise is only concerned with one part of 

a very extensive subject. The modern theory of Partitions will 

„ be found in the first part of the article by MacMahon mentioned 

« above. 

V The first six chapters — a great portion of which, we hope, will 
%- be found easy reading — may be said to lead step by step to 
N^j Gordan's wonderful proof of the finiteness of the system for a 
single binary form. The sixth chapter is, in fact, devoted to an 
exposition of Gordan's thii-d proof, but here, as throughout the 
book, we have allowed ourselves a free hand in dealing with the 
memoirs and treatises quoted. For example, we have made much 
use of Jordan's great memoirs on Invariants in proving Gordan's 
theorem: in a later chapter on Types of Co variants the development 
of Jordan's method has led us to some results which we believe 



2609G3 



VI PREFACE 

to be important as well as novel, notably to an exact formula for 
the maximum order of an irreducible covariant of a system of 
binary forms. 

The remainder of the book is mainly of geometrical interest : 
much space is devoted to Apolarity and Rational Curves, and the 
treatment of ternary forms is from the geometrical rather than 
the analytical point of view. The only complete system of 
ternary forms given is that for two Quadratics: it may be felt 
that more should have been said on this subject, but we think that 
with the methods known up to the present the treatment of 
ternary forms is too tedious for a text-book. 

The number of references to Mathematical Journals etc. will 
perhaps be found unusually small : for this there is no need to 
apologise since the admirable Bericht ilber den gegenwdrtigen 
Stand der Invariantentheone* of Meyer gives references up to 
the last few years and in a more complete fashion than is 
desirable in a book which makes no pretensions to being exhaustive. 

We wish to thank Dr H. F. Baker for help given to us in our 
early reading and Professor Forsyth for encouragement while 
writing. For reading of proof-sheets we are indebted to Mr J. 
E. Wright, B.A., of Trinity College, Mr P. W. Wood, B.A., of 
Emmanuel College, and in a still greater degree to the late 
Mr A. P. Thompson, B.A., of Pembroke College, whose Enthusiasm 
for Mathematics and research was most helpful and whose early 
death is deplored alike by his teachers and his fellow-workei*s. 
Our thanks are also due to the officials of the University Press 
for great help received during the course of printing. 

J. H. GRACE. 
A. YOUNG. 

* Jahresbcricht der Deutschen Mathematiker Vereinitfunrj, Vol, i., 1892. French 
translation byFehr; Gauthier-Villars, Paris, 1897. Italian translation by Vivanti; 
Pellerano, Naples, 1899. Article, Invariantentheorie in the Encyclopadie der 
iiMthcmatischen Winsenschaftcn. 

August 18, 1903. 



CONTENTS. 



CHAP. 
I. 


Introduction 










PAQE 

1 


II. 


The Fundamental Theorem 










21 


III. 


Transvectants .... 










36 


IV. 


Transvectants {continued) 










53 


V. 


Elementary Complete Systems 










85 


VI. 


Gordan's Theorem .... 










101 


VII. 


The Qcintic 










128 


VIII. 


Simultaneous Systems . 










158 


IX. 


Hilbert's Theorem 










169 


X. 


Geometry 










183 


XI. 


Apolarity and Rational Curves 










213 


XIT. 


Ternary Forms .... 










246 


XIII. 


Ternary Forms (coniinued) . 










274 


XIV. 


Apolarity (contimied) 










299 


XV. 


Types of Covariants 










319 


XVI. 


General Theorems on Quantics . 










339 


Appendix I. The Symbolical Notation 










365 


)j 


II. Wronski's Theorem 










370 


>» 


III. Jordan's Lemma 










375 


Ii 


IV. Types of Covariants 
sidex 










. 378 
. 381 



4 



CHAPTER I 

INTRODUCTION. SYMBOLICAL NOTATION. 

1. If in the expression 

we write Xi = f^Xi + t/i X2 , 

W2= ^2X1 + 7)2X2, 
we obtain a new expression, viz. 

AoXi' + 2A,X,X2 + A.,X2^ 
where ^0 = flo ^1^ + ^(h ^1 f 2 + ^2 ^2', 

-4i = ao|i»7i + «! (|i7;2 + |2'7i) + a2^2'72, 
A2 = fto'/i^ + '^(hViV2 + (hvi- 
It is easy to verify the identity 

A^A^-A^'^ (aoUo - ai") (li?/, - ^^Vif, 

which shews that the function AqAs — A^^ of the coefficients of the 
transformed expression differs from the same function ao<*2 — ^i" of 
the coefficients of the original expression by a factor involving 
only the coefficients contained in the transformation. 

2. In the present work we shall give an account of the 
theory and structure of functions of the coefficients possessing 
properties analogous to that described above ; but before pro- 
ceeding to generalities we shall give some further examples. 

If we transform the two expressions 

^01 •" ^Otiwju/2 "T" CLnOCi^ f 

(^0 ^1 I ^(l\ X-yX^ T 0-2 X2 ) 

G. & Y. 1 



2 . * THE ALGEBRA OF INVARIANTS [CH. I 

in the same way as before, and they become 

A;Xi^-\-2A,'X,X, + A,'X,\ 

then it is easy to verify the identity 

AoA^' - 2A, A,' + Ao'A^ = {a^a^ -2a^a^'-\- a^a^) (^^^2 - ^^ViY- 

Thus we have here a function of the coefficients of two ex- 
pressions such that the new value differs from the original value 
by a factor depending only on the transformation employed 

3. As a third example, if the cubic expression 

become A,X^' + ;Mx X;'X^ + ^A^X, Xi + A^Xi, 

when we put a\ = l^jXi + t^iXj, 

then we have 

(^0^2 - A^) X^ + (^0^3 - ^1^2) ^1X3 + (^1^3 - A^) Z/ 

This identity indicates a property quite similar to that illus- 
trated in the two previous examples, but the function, which is 
unaltered except for the factor (^i?72 — ^2'7i)^ now involves the 
variables as well as the coefficients of the expression from which 
it is formed. 

The result we have written down may be verified directly, 
but more easily as follows : 

Denoting the original expression by f and the transformed 
expression by F we have to prove that 

aAvaZa* \dx,dxj \dx,^dx^^ Kdx^dxJ]^^'^' ^'^'^^ 

dF__dF_dx^ dF_dx^ 
dXi dxi dXi dx^ dXi 

'^'dx^'^^^dx^' 



2-4] INTRODUCTION. SYMBOLICAL NOTATION 3 

and in like manner 

d^F d^F d^F d^F 

d^F d^F d^F d^F 

d^F ,d'F _ d^F ?'F 

But these equations are exactly the same as those which 
express A^, A^, A2 in terms of Uo, a^, a^ (§ 1), hence 

The expression ^^ ^7^ — 5~ra ) ^^ called the Hessian of/. 

4. Let us now explain the phraseology in common use when 
dealing with questions such as arise in our subject. 

Quantics. A rational integral homogeneous algebraic function 
of any number of variables Xj, x^, ... x^y is called a quantic. 

The degree in the variables is called the order of the quantic, 

and according as the number of variables is two, three, four 

v.'e call the quantic binary, ternary, quaternary 

Thus a binary quantic of order w is a rational integral 
homogeneous algebraic function of two variables which is of the 
nth degree in those variables. 

Such a quantic might be written 
but we shall find it invariably more convenient to write it 

i.e. with binomial coefficients prefixed to the various a's. 

The former of these expressions is now commonly written 

Vflo> ^1) *2> •••» (^nyj^\i '^2) > 

and the latter {a^, a^, a^, ..., an\xi,x^^, 

a very convenient notation introduced by Cayley. 

1—2 



4 THE ALGEBRA OF INVARIANTS [CH. I 

The mere consideration of the transformation of the binary 
form 

will be sufficient to convince the reader of the advantage of the 
introduction of binomial coefficients. 

Passing now to the case of any number of variables, we call 
the quantic a p-ary ^-ic when it is homogeneous and of degree q 
in /) variables. 

Thus the most general ternary quadratic is written 
and in general the ternary n-ic is written 

where the summation is extended to all values of j9, q, r satisfying 
the equality 

p + q + r = n. 

It will be noticed that here we have prefixed multinomial 
coefficients to the as. 

5. Linear Transformations. The equations 

X.i = ^2^1 + '72>X^2 

are said to constitute a linear transformation from the variables 
x^x^ to the variables X^X^ — it is of course implied that the 
coefficients on the right do not involve either set of variables. 

The determinant 

is called the determinant of the transformation. 

If D vanishes it is evident that x^ and x^ are virtually identical, 
for their ratio is constant, and hence, as the variables are always 
supposed to be independent, we shall throughout only deal with 
transformations which have a non-vanishing determinant. 

On solving for X^X^ we find 

Xi = {7)^Xi-'r],X2)/D 

* 

Xi = {-^2Xi + ^iXi)II> 



4-6] INTRODUCTION. SYMBOLICAL NOTATION 6 

SO that the passage back from the new variables to the old is 
eflfected by a linear transformation. This is called the inverse 
of the original transformation ; it is evident at once that its 

determinant is equal to ^ . 

6. Let us now regard a linear transformation as an operator, 
which acting on a;^, x^ changes them to X^, X^, and let us consider 
the effect of two such operators acting successively. 

If the coefficients of the first are 
and those of the second 

y I ' . f- / / 

?1 , ''71 , ?2 > '72 , 

then we have 

a^i = ^i^i + »7i^2) 

^2=^2^1 + »?2-X'2J ' 

-X'l=^l'^l' + ^l'Z2'| 
X2=^2X^ ■\-'q^X^\ ' 

and the effect of the two operators acting successively is to change 
from the variables cci, x^ to X/, X^. 

Now on elimination of X^ , X^ we find 

^i = (111/ + % la') X^ + (Iit;/ + -^iT?/) X^, 

^2 = (I2I1' + ^7212') ^1 + (hvi + V2V2) X^'. 

And accordingly we can pass directly from the original to the 

final variables by means of a single linear transformation which 

we shall call 2. 

If we call the two preceding operators S and 8' we may write 

X = SS' 

and 2 is called the product or the resultant of S and S'. 

It must be carefully noticed that the order of the factors 
S and S' is essential in considering their product. In our 
example we supposed that S acted first and then S'. If S' had 
acted first and then S we should have 

X = S'S 

and it is manifest that S and S' are not in general the same. 



6 THE ALGEBRA OF INVARIANTS [CH. I 

Since the resultant of two or any number of linear trans- 
formations is another such transformation, the whole set of linear 
transformations obtained by varying the coefficients is said to 
form a group — a continuous group because the coefficients ^ and 
17 may be sxipposed to vary continuously. 

The determinant of 2 is equal to the product of the determi- 
nants of S and 8', as follows from the multiplication theorem for 
determinants. 

The product of a transformation and its inverse is a trans- 
formation which does not affect the variables, i.e. it is 

which is called the identical operator. The determinant of this 
is unity, and, as we have pointed out, the product of the deter- 
minants of a transformation and its inverse is also unity. 

7. The idea of a linear transformation admits of immediate 
extension to any number of variables x^, Xi,...Xp and now the 
transformation consists of n equations 

Xr='^nX-^-\-^nX^+ ... + ^rpXj,, r=\, 1,...p. 

The determinant D formed with the |'s for elements is called 
the determinant of the transformation, and inasmuch as when D 
vanishes there is a linear homogeneous relation between the x's, 
we exclude as before all transformations having a vanishing 
determinant. 

If Z) 4= we can solve for the X's in terms of the x& and, as 
can be easily seen, each X is a linear function of x^, x^, ... Xp, so 
that we have 

Xr = Vn^i + Vr2^2 + • • • + Vrp^p> 

a linear transformation which is the inverse of the preceding one. 

As in the case of two variables, the resultant of two linear 
transformations S and T is a third linear transformation 

and on examining the coefficients in S it will be seen at once by 
the multiplication theorem that the determinant of % is the 
product of the determinants of /S and T. 



6-10] INTRODUCTION. SYMBOLICAL NOTATION 7 

8. In the earlier portion of this work we shall deal almost 
entirely with binary forms, and although we shall be constantly 
considering linear transformations and their etfects, yet the fact 
that they form a group will not be explicitly used. Our only 
object, in introducing these elementary properties of groups, is to 
point out that the connection between invariants and groups is 
intimate and universal — in other words, that every group has its 
accompanying invariants and, conversely, every set of invariants 
belongs to a group. 

9. Invariants of Binary Forms. If a binary form / 
be changed by a linear transformation into a new form F, and a 
function / of the coefficients of F be equal to the same function 
of the coefficients of / multiplied by a factor depending solely on 
the transformation, then I is called an invariant of the binary form/ 

Thus for example in § 1 the identity 

(^0^2 - ^i') = (ao«2 - ai) (^iV2 - ^2Vi)" 
shews that a^a.^ — a-^ is an invariant of the binary quadratic 

An exactly similar definition applies to a joint invariant of 
several binary forms, e.g. 

a^a^ — ^a^tti + a^a^ 

is an invariant of the two binary forms 

and adx^ + '2a-lx■^X2 + a^x^. 

10. For the present we shall confine our attention to invariants 
which are rational integral functions of the coefficients. It is easy 
to see that there is no further loss of generality if we suppose the 
invariants to be homogeneous in each set of coefficients that they 
contain. 

Thus for example if / be an invariant of a single binary form 
f which is not homogeneous in the coefficients a we can write / in 
the form /i + /a + . . . + /«, 

where each element in this sum is homogeneous. 



8 THE ALGEBRA OF INVARIANTS [CH. I 

Now by definition we have 

I{A) = MxI(a), 
and therefore 

I,{A) + I,{A) + ... + Is(A) = M{I,(a)+I,(a) + ...+I,(a)}. 

But the A's are linear functions of the as and M is independent 
of both, and therefore the only part on the left-hand side which is 
of the same degree as I^ (a) on the right-hand side is Ii{A); 

.'. h{A) = Mh{a), 

that is to say I^ is an invariant. Hence a non-homogeneous 
invariant is the sum of several homogeneous invariants. 

This result can be at once extended to any number of binary 
forms. 

As an example 

ttoOs — a^ + a^a^ — 2a^a/ -I- a^ao 
is an invariant of the two binary quadratics 

(tto, Oi, as 5^1, ^2? and (ao', a/, a^'^Xi, x^f, 
but it is the sum of two expressions 

ttotta — tti", 

and aoOa' — 2aia/ -I- a<iao 

each of which is homogeneous in the two sets of coefficients. 

11. Covariants of Binary Forms. If a binary form / is 
changed into a form i'^ by a linear transformation, and a function 
C of the coefficients of F and the new variables Xi, X^ be equal to 
the same function of the coefficients of /and the old variables ajj, a;, 
multiplied by a factor depending only on the transformation, then 
C is called a covariant of the binary form. 

Thus from what we have seen 

ay ay ( df Y 

dxi^ dx^ \dxi dx^ j 

is a covariant of the binary cubic/ and in fact of any binary form. 

An exactly similar definition applies to a joint covariant of 
several binary forms — as an example the reader will have no 
difficulty in shewing that the Jacobian 

§/;a5^_a/a0 

dxi dx^ dx^ dxi 



10-13] INTRODUCTION. SYMBOLICAL NOTATION 9 

of any two forms / and ^ is a covariant of those forms, the 
multiplier being (^1% — |^2^i) the determinant of the transformation. 

We shall confine our attention to covariants which are rational 
integral functions both of the coefficients and the variables, and, as 
in the case of invariants, there is no difficulty in seeing that there 
is no further loss of generality in supposing such covariants to be 
homogeneous in the variables and in each set of coefficients involved. 
In fact if a covariant be not homogeneous it is the sum of several 
parts each of which is a covariant and homogeneous. 

12. Degree and Order of a Covariant. The degree of a 
covariant of a single form is its degree in the coefficients of that 
form — the order is the degree in the variables. 

The covariant -~^ ^^ — f ^ ^ ) of a binary form of order n is 

of degree two and order 2n — 4. 

A covariant of several binary forms has a definite partial degree 
in each set of coefficients involved and the order is as before the 
degree in the variables. 

The Jacobian of / and <^ is of degree one in the coefficients of 
each of the two forms, and its order is the sum of the orders of 
f and (f) diminished by two. 

13. Symbolicar Notation. In our investigations we shall 
find it of the utmost value to write the binary quantic 

ttoXi^ + naiXi^~^ 0)2+ ... +{ j a^a^i""'' a;/ + . . . + a^iPa" 
in the symbolical form 

so that tti" = tto, tti"""^ a.2 = ttj , . . . a^"-'' a/ = ar,... ^^ = an. 

This representation is startling at first sight, but consider how the 
use of it would introduce errors into calculation. They would 
arise because relations of the type 



ana» = a 



2?l— 2 ~ 2 /, 2 



between the coefficients prevent our binary form from being a 
general one. Now in representing a function of the coefficients 



10 THE ALGEBRA OF INVARIANTS [CH. I 

symbolically we allow no symbol such as a to occur more than n 
times in any one term, so that the possibility of relations giving 
rise to 

aotta = O]^ 

is entirely precluded. In fact to orbtain this relation there must 
be 2n a's multiplied together in the representation of the function 
aorta or al^ whereas, when we allow no more than n a's to occur in 
any one term, the (n+1) expressions 

aA ai"-ia2,...ai"-'-a2'-, ...aa" 

are independent quantities, i.e. with these restrictions on the use 
of our symbols the {n + 1) coefficients of the original quantic are 
not necessarily connected by any relation, and therefore the most 
general quantic can be represented in the form indicated. 

Accordingly in addition to the symbol a we introduce a 
riumber of equivalent symbols /3, 7, ... so that 

/= (aia^i + a^x^)'^ = (^^x^ + ^^x^Y = (y^x^ + y^x^Y =... 

or as it will invariably be written 

J — Clx—Px—'Yx ••• 

The symbolical equivalent of aorta is not 

because here there are more than n a's multiplied together. 

To represent rtort^ we must use two different symbols a, yS and 
then 

aoaa = ai"/8i"-^A'> 
which is of course equivalent to 

/9l''a,"-2a2^ 
whereas in the same symbols a,'' is represented by oii^~^okfii^~^^2' 

In general to represent an expression of degree m in the 
coefficients, we have to use m different symbols of the type 
a, A 7. •••• 

We have said that not more than n a's must be multiplied 
together in a given term — on the other hand if the expression has 
an actual as well as a symbolical significance not less than n of 
these symbols must occur together because only the expressions 

ai", tti'^-^aa, ... Oa'' 
have an actual meaning. 



13-14] INTRODUCTION. SYMBOLICAL NOTATION 11 

14. A function of the coefficients can generally be represented 
symbolically in different ways as we have seen in the case of aoa, 
for example, which is equivalent to both 

an^n-2^2 and ^^^a,-^-^. 

There is one method of determining the symbolical repre- 
sentation which is very convenient because it often leads to the 
expression most suitable for our purpose. 

Suppose, in fact, that P is a homogeneous function of the mih 
degree in Uq, a^, ... an, then 

is only of degree m— 1 in a^, ai, ... an. If in Pi we replace each 
b by the corresponding a we obtain mP, as follows from Euler's 
Theorem relating to homogeneous functions. 

In like manner if in 



A i_Vfo~4-&— +b -)P 

^9ai "" '^dajx'^dao ^ dai '" ^daj 



we replace each c and each b by the corresponding a we get 
m (m — 1) P and P^ is of degree m — 2 in the a's. 

Proceeding in this way we can find an expression Pm-i which 
is linear in each of m sets of symbols 

a, b, c, ... k, 

and which becomes equal to P x ml when each b, c, ... k is 
replaced by the corresponding a. 

Now having formed the expression Pm-i we replace each a 
by the symbol a, each b by the symbol yS, each c by the symbol 7 
and so on. Since the expression is linear in each set of letters, 
each symbol will occur exactly n times in every term, and then, 
regarding the symbols as referring to the same quantic, we have 
the required symbolical expression. 

Thus for example 

aoa, - a,^ = i (b, ^ + b, ^ + 6, Aj («„«^ _ a,%^^ 

= i (&o«2 + &2«o - 2ai6i)j=a 

= i (AW + ^,W - 2aia,,8i A) a,-'^,^-' 



12 THE ALGEBRA OF INVARIANTS [CH. I 

and the convenience of this expression in terms of a, y8 will be 
abundantly evident in the sequel. 

Ex. (i). For the binary quartic shew that 

and flo «i «2 | = H«i/32-«2/3i)^Oiy2-^2yi)^(yia2-y2«i)*- 

Ui a^ ct^ I 

ttg CTg ^4 I 

Ex. (ii). By the same method shew that for any binary form 

Ex. (iii). Shew that for a binary form of odd order (a^^2 - cii&i)"' is zero 
and write down its value for a form of even order in terms of the coefficients. 



15. Polar Forms. The expression 
(n-r)\f d d 



y^^-^y-' 



-Y/ 



where y*= aa;" = )Sa;'' = etc. 

is a binary form of order n, is called the rth polar of/ with respect 
toy. 

The operator ( t/i ^ — \-y^K-\, which is frequently written 

l^'-a^. + ^'a^J-^ 

is said to be derived from / by polarizing r times with respect 
to y. The numerical factor ^^ -^ is only introduced for con- 
venience. 

These polar forms admit of very simple representation in our 
symbols, for 

and so on. 

Hence the rth polar of / with respect to y is 
(n — r)! 



w! 



w (71 - 1) . . . (n - r + 1) (x^-''tLy\ 



that is OiJ^-^'aJ. 



14-16] INTRODUCTION. SYMBOLICAL NOTATION 13 

The differential coefficients of f with respect to the variables 
are particular cases of polar forms. 

For if 2/1 = 1, 2/2 = 0, the rth polar is 

ay 

aa;/ 
and if y-^ = 0, 3/2 = 1, the rth polar is 

9^2""' 
In general we have 

aP+«/" n ! 

J^ — /y n-p—qn Pn 9 

^x^-dxS {n-p-q)\ "" ' '• 

The form ^ ^ [y^] (•^5~)/^^ called a mixed polar 

with respect to y and 2 ; its symbolical expression is 

a^n-p-q^a^q^ 

16. Effect of a Linear Transformation. If we write 
^^1 = ^1^1 4- 771X2 

^2 = ^2X1 + Vi^z, 
then Ux becomes 

or ajZi + a^Xg, 

and hence the binary form a^^ becomes 

(afZi + a,Z2)" 

or ai" Xi" + n aj'*-^ a, Xi"-^ X^+...+ a^X^. 

Accordingly in the transformed expression the coefficient of 
Xj** is found by replacing x by ^ in the original form, and the 
coefficients of Xi'*~^X2, X-^~'^X} ... are found by polarizing the 

coefficient of X-^ with respect to i) once, twice Of course 

suitable numerical multipliers must be introduced. 

The reader will easily illustrate this result by reference to the 
transformation of a binary quadratic in § 1. 



14 



THE ALGEBRA OF INVARIANTS 



[CH. I 



= 


«!, 


"2 


X ^1, 


^. 




^1, 


A 


Vi, 


V2 



17. The form a^^ = ^x^ = 7a;" • . . becomes on transformation 
(a^X, + a,X,r = (^f Z, + /3,X,r = (y^X, + y.X.y =.... 

Now we have 

All + Af2, /3l»7l + A'?2 

= («! A - otsA) (^i'72 - fa'/i), 
a result of fundamental importance. 

We shall denote the expression (ai/Sa — tto/Si), which we call a 
symbolical determinantal factor, by (a/3), so that (ay3) = — (/9a), and 
the above relation may be written 

af/g,-a,;8i = (a/3)(^^). 

To illustrate these remarks let us prove that 

do 0^4 — 4aia3+ Soa^ 

is an invariant of the binary quartic 

We have 

(00^4 - ^cb\(h + 302=*) = :| f 26 5- j (ttotti - 4aia3 4- ^a^)h=a = i (a/8)*. 

Thus, if the coefficients of the new form be denoted by capital 
letters as usual, we have 

(^0^4 - 4^i^3 + 3^2^) = i (aj A - a, A)* 

as follows from the symbolical expression given above. 

But since a^ ^, - a,ySf = (a/9) (^17), 

A^A^ - 4^i^3 + 3^2^ = {^tjY (aoa^ - ^a^a^ + ^(h% 

which shews that a^ai— 4iaiaz + Za^ is an invariant and that 
the multiplying factor is the fourth power of the determinant 
of transformation. 

18. Symbolical expressions representing Invariants. 

If the symbolical equivalent of an expression /, homogeneous and 
of degree % in the coefficients of the binary form 

be an aggregate of terms each of which is a product of factors of 
the type (ay6), then / is an invariant of the quantic. 



17-19] INTRODUCTION. SYMBOLICAL NOTATION 15 

For let I = ^T, where T is the product of w factors of the 
type (ajS), then the total degree of T in the symbols is 2w and 
it is also n x i, for there must be i sets of symbols, each set 
occurring to degree n\ therefore ni=2w, so that w is the same 
for every term in the aggregate representing /. 

If /' be the same function of the coefficients of the transformed 
expression, then /' = ^T' 

where T' is found from T by replacing a^ by a^ , a.2 by a, and so on. 

But since (a^/S, - a,/3f) = (a/S) {^t}) 

it follows at once that T = {^y T, 

and therefore I' = {^7))^ I since w is the .same for every terra. 

Hence / is an invariant. 

Exactly the same result is true for any number of binary 
forms if we suppose that / is homogeneous in each set of 
coefficients, for it is easily seen that the number of determinantal 
factors must be the same in every term, it being in fact 

when Til, n^, ... are the orders of the forms and i^, i^, ... the 
respective degrees of / in the coefficients of the forms. 

The rest of the proof then depends only on the fact that, 
whatever a and /3 are, we have 

Thus / is an invariant and the multiplying factor is now 

19. This simple theorem enables us to construct as many 
invariants as we please — we have only to write down a product 
of factors {a^) and take care that the symbol a occurs in n of 
these factors where n is the order of the form to which the symbol 
a belongs. If this condition be not satisfied the invariant 
property still holds but the expression has only a symbolical 
meaning. On the other hand, if every symbol occur to the right 
degree but the expression be not reducible to the form above, it 
is an actual function of the coefficients which is not an invariant. 



16 THE ALGEBRA OF INVARIANTS [CH. I 

As an example we have an invariant of the second degree 
(fliS)" for a binary form of order n. This vanishes identically 
when n is odd, as can be seen by expressing it in terms of the' 
coefficients ; or thus, since a, /3 are equivalent symbols 

and hence (a^)" = (-l)"(ayS)", 

giving the result at once. 

Again, for the binary cubic we have the invariant 

and for the binary quartic the invariants 

(a/3)^ (a^n^yYiyay, {a^y(ay)(^8){y8y. 

In every case it will be observed that the multiplying factor is 
a power of {^v)- 

As an example of invariants of several binary forms we may 
mention (ayS)", an invariant of the two different binary forms 
fla:" and /9a;". For quadratics this is the well-known invariant 
of §2. 

Again (a/3) is an invariant of the two linear forms a^ and ^^ 
and in this case a, /8 are actual coefficients as well as symbols. 
Then (ayS) (ay) is an invariant of the quadratic Ox^ and the linear 
forms /3a,, y^. 

20. Covariants. A similar method exists for constructing 
covariants. 

Commencing with an example let us prove that the Hessian 

^_ayay /ayy 

dxi^ dxi \dxidxj 
is a covariant of the binary form /= a^^ = /3a;" = . . . 
Since H is of the second degree in the coefficients 

^ V daJ Ida;,' dx^^ \dx,dxj j b=a 

\dxi^ dx^ "dx^ dx^ dx^dx^ dx^dxj 6=a 
where /' = {\h^ . . . bn$XiX^y\ 



19-20] INTRODUCTION. SYMBOLICAL NOTATION 17 

Replacing the as by as and the 6's by /3's as usual, we have 

H=^n'' (n - 1)2 {ai^a^'^-^ye^^yga.'^-a 

+ a.^a,-^ A^^^-2 - 2a,ct,a^"-^ A/82/3«,"-='j 

as can be immediately verified by expressing this in terms of the 
coefficients. 

The transformed quantic is 

and the corresponding expression derived from this is 

(a^/3, - a,/3|y^ (ajZ, + a^X^y-'i^^X, + ^,X,f-' 

which shews that the expression is a covariant and that the 
multiplying factor is (^■j/)^ 

In general, if an expression C, of degree i in the coefficients 
of/ and of order m in the variables, can be symbolically repre- 
sented as an aggregate of terms, each of which is the product 
of a number of factors of the type (a/S) and a number of the type 
a-t, then G is a covariant oif. 

In fact let C = SF, where F is such a product. 

The number of factors with suffix x in F must be m, the order 
of G, and if w be the number of the type (a/8) we have 

2w + m = ni, 

for each of these represents the degree of G in the symbols. 
Hence lo is the same for every term. 

If C" be the corresponding expression derived from the 
transformed quantic, then 

G' = 2F', 

where F' is derived from F by replacing «! by a^, a^ by a, and 
so on, and «« by {a^X^ + a^Xg). 

Thus since (a^^, - a,/3f) = (a/3) (^t;) 

and a^ = a^Xj + a^Xj 

we have F = {^r}Y F, 

.-. C" = (|^)«'C, 

that is to say C is a covariant. 

G, & Y. 2 



18 THE ALGEBRA OF INVARIANTS [CH. I 

Exactly the same method applies to a covariant of any number 
of binary forms, but now the symbols a, /S, ... may refer to different 
forms and, of course, a symbol such as a. must occur in the symbolical 
expression to the requisite degree. 

We can thus easily construct any number of covariants of one 
or more forms, e.g. for a binary form of order n 

is a covariant for any integral value of r, but it vanishes when r 
is odd. 

Again, if <x^, ^^ are two different quantics, 

is a covariant ; if r = 1 it is the Jacobian. 

As further examples we have the covariants 

(a/3)» {&if ina)- ct,;S,7„ (ay8) {^i) (7a) (aS) (^g) (7S) a^'^i^A^ 

of the binary quintic 

a.'' = y8.' = 7^» = a/=... 

As an exercise the reader may prove that the last one 
vanishes identically. 

21. We have seen how useful the symbolical methods are in 
constructing invariants and covariants. In the next chapter we 
shall prove that they constitute an ideal calculus when we shew 
that every invariant and covariant can be represented as a sum of 
symbolical products of factors of the types (a/S) and a^. Meanwhile 
anticipating this result we shall indicate the methods of trans- 
forming symbolical expressions. These depend on two principles : 

(i) Interchange of equivalent symbols, 

(ii) Identities in symbolical expressions. 

According to (i) if a symbolical expression have an actual 
meaning and contain two equivalent symbols then its value is 
not altered by interchanging those symbols. We have already 
used this method in proving that the invariant 

(ayS)" of the quantic a^^ = /3^" 



20-22] INTRODUCTION. SYMBOLICAL NOTATION 19 

vanishes when n is odd. As another easy example we have 

(ay8)(/37)(7a) = 
for the quadratic a^^ = jSx^ = yj', 
or for the two different quadratics a^^ = ^^^ and yx^. 
More generally the co variant 

(ayS) (M (7^) ax''^^x''-V-' 

is alw^ays zero unless the three forms a^^, /3a;" and jx^ are all 
different. 

22. Fundamental Identities. We have identically 

(Moix + (ya)^x + {ci^)yx==0 (I). 

as can easily be verified. 

From this identity many others may be deduced. 
For example, replacing cc^ by B2 and x^ by — Si we have 

(/37)(aS) + (7a)(;S8) + (a/3)(7S) = (II), 

a result useful in transforming invariants. 
Again from (i) 

(M ^x = (^a) Ix - (7«) ^a; 
and hence by squaring 

2 (otyS) {ay) ^xlx = (a/3)-^ Ix' + {ayf ^x' - {^yf a,^ • • .(HI). 
As identities less generally used we may mention 

{^yf ax' + (7a)' ^x' + {oi^r Ix' = 3 (M (7a) (^'/S) aa=/3a=7* 

()S7)'«.^ + (7«)^/Sa.^ + («;Sy7*' 

= 2 {(a;8)^ (a7)-^ /3,.^7,,^ + {^yf {^af yx^ai + (7«)' (7/^)^ a*^/3«=1. 

Ex. (i). For the quadratic 

(a^) (ay) /3,y, = ^ {(a^)^ y^ + (ay)^ ^,2 - {^yf a,^} 

since the symbols a, /3, y are equivalent. 

2—2 



20 THE ALGEBRA OF INVARIANTS [CH. I 

Ex. (ii). If fi = a^=^^ and fi — ^x=^x be two different quadratics, 
to express the square of the Jacobian J={aa)axa'x in terms of/ and /'. 

We have J^={aa') a^a'^ {m') ^x^'x 

or since (|3/3') a'^={^a') ^'^ - O'a') ^^ 

J^={aa') i^a') a,^,^J^- {aa') O'a') a,^'^ . /3,2 

= i3',2 i {{aa'f ^,2 + Oa')2 a,^ - {a^f a',^} 

-^,m(aa'r^,^ + {P'ayaJ-{a^ya'J^} by (III), 

or if (a/3)2=/,i, (aa')2 = (a0')^=...=/,2, {a'^f = l^^, 

we have 2 J2 =/2 {/j^^/-^ + I^J^ _ /^^/J 

~/i{m2/2 + A2/1 ~ ■'12/2} 

Ex. (iii). Prove that for the binary quartic 

(a^)2(ay)2^,2y^2=.|_^.(„^)4 
(o^) (ay) a,2/3,3y^33=^y. („^)2 „^2^^2. 



CHAPTER 11. 

THE FUNDAMENTAL THEOREM. 

23. It will be remarked that in every example of in- 
variants and covariants, discussed in the preceding chapter, the 
symbolical expression for such a function involved only factors 
of the types (a/3) and a^;, and further that the multiplier alluded 
to in the definition was always a power of the determinant of the 
transformation. We are now going to establish the general truth 
of these properties. 

As a matter of history, we may observe that the original 
definition of an invariant stated that the multiplier was of the 
form mentioned ; but following the logical, rather than the 
historical order, we shall first prove that the multiplier must 
be a power of the determinant and then proceed to prove the 
proposition relating to the symbolical forms for invariants and 
covariants. 

24. Suppose that / is an invariant or covariant of a single 
binary form f — after what has been said, § 10, we may assume 
that I is homogeneous in the coefficients of/. 

Let the linear transformation 

change / into /' and let /' be formed from /' in the same way 
that / is formed from f\ then, by definition, 

/'=i^(^i,77i, f 2,772) x/ 

and we have to shew that F is simply a power of (f 1772 — ^2''7i)- 



22 THE ALGEBRA OF INVARIANTS [CH. II 

Now let a second transformation 

Xi ^= q\Xi + 7Ji X^ 
X^ ^ 52 '^\ "■" '72 ''^2 

change /' into /", and let /" be formed in the same way from f", 
so that 

Hence we have 

/" = ^(ri. '71. ^2, 1.) X F{^,\ 7;/, ^/, ,;;) X /. 

But we can pass from the variables x^, x.^ to the variables 
x", x^' by the single transformation 

Xi = (fi^i' + 171I2') a;," + (^i77i' + T/iW) a^a" 

a^2 = (^2^1' + '72I2') ^1" + (^2';i' + '72'?2') ^2" ; 

therefore 

Consequently F must satisfy the functional equation 

= F(^„ 71,, ^2, %) X F{^^, V. r;, ^72'). 
The solution of this equation is not difficult. In the first place 
we remark that since ^1 = 1, 7/1 = 0, ^2 = 0, % = 1 gives the 
identical transformation, 

F{\,0,i),l) = \. 

Again putting ^1 = k, 7/1 = 0, I2 = 0, '72 = ": each new coefficient 
is equal to the corresponding original coefficient multiplied by 
the same power of k, in this case the multiplier is clearly a power 

of K, i.e. 

F{k, 0, 0, /«;) = «:'■. 
Since 

F{^u Vi,h, V2) X F(k, 0, 0, k) = F(k^„ KVi, K^2, 'CV2), 
we have 

F(k^i, /CT/i, K^2, icVt) = K-^'Fi^i, 77i, ^2. '72), 

therefore F is homogeneous and of degree r in the four variables 
1^1. ^71. 1^21 Vi- 

Finally let us choose ^Z, t;/, I2'. %' so that 

^i|i' + 771^2' = 1 > |i»7i' + »7i'72' = 0. 
^2^^/ + ^72^2' = 0, |2'7i' + '72^2' = 1. 



^B 94-_9fi1 



24-26] THE FUNDAMENTAL THEOREM 23 

which relations give 






{D = ^j772 - ^27/3), 



then we have 

Fi^,, Vl, ^2, V2) X i^(^/, %', f/, 772') = i^(l, 0. 0, 1) 

Consequently since ^ is homogeneous and of degree r 

F (^1 , %, I2, '^a) X i' (772, -vi, - ?2, f ) = ^. 

But inasmuch as D is obviously irreducible — i.e. it cannot be 
resolved into factors — and F is clearly an integral function, this 
equation shews at once that both 

-^(^1. Viy ^2, V2) and F(r}2, - r)^, - ^2, ^1) 

are integral powers of D. 

Hence the theorem is established. 

25. Assuming the truth of the proposition just proved, the 
proof of the fundamental theorem that invariants and covariants 
can be completely represented by factors of the types (a/8) and a^ 
is very simple in principle. The actual work requires two lemmas 
of great importance in the present subject, and we shall give them 
separately. Tbey are both concerned with properties of the 
differential operator 

26. Lemma I. If n be a positive integer 



In fact ^— (x^y^ - x^y^Y = ny^ {x^y^ - x<^iT~^, 

+ ?? (n - 1 ) (^13/2 - iio.jy^y-^'X{y^ 



24 THE ALGEBRA OF INVARIANTS [CH. II 

Similarly 

92 
^— - {x^y^ - x^,y =-n {x^y^ - x.y^f-^ ^n{n-V) (x^y^ - x^y.f-^x^y, . 

Consequently 

n (^i2/2 - x^y^Y ={n(n-l) + 2n} {x,y., - x^y.y-' 
= n{n+l) {x^y., - x^yif-\ 
which establishes the lemma. 

If we operate again with H we find 

fi2 (^x^y^ - x^y^Y = n{n + \){n-\)n{xiy^- x^y^f-'' 
and in general 

= (n + l)n'{n-iy ...{n-r + 2)^ {n-r + 1) {oc^y^-x^y-,Y-^ 
or n*- {xyY = (n + 1) /i^ {n-iy ... {n-r + 2)^ {n-r + 1) {xyy^. 
Finally 

n« (a^i 2/2 - ^22/1)" = (n + 1) (w n^' 

a constant which is not zero — for our immediate purpose this is 
the important result, and it can be at once verified by expanding 

\dx,dy~ dx,dyj ' ^^^2/2-^2^1) 
by the Binomial Theorem. 

27. Lemma II. If the operator fl be applied r times to the 
product of m factors of the type a^. by n factors of the type fiy, 
then each term in the resulting expression contains r deter- 
minantal factors {ol^), (m — r) factors Ox and {n — r) factors ^y. 

To ensure perfect generality we consider 

where P = Oa,*" aa,<''' . . . a^;*'"' 

and Q = /3/)^/'.../3,<»', 

the a's and the ^'s being all different. 



H 26-28] 
Wm Now 



26-28] THE FUNDAMENTAL THEOREM 25 



where the summation extends so that r takes all the values 
1, 2, ... m and s takes all the values 1, 2, ... n. 



Hence on subtraction 

P Q 



nP.Q = SK)/3'«0.,,,^„. 



which establishes the lemma for r = 1. 

But since the operator CI has no effect on a factor of the 
type (a"''yS<**) the theorem holds for r = 2 ; in fact 



and performing the operation on the right we have the result. 

Proceeding in this way we see that at each step a new factor 
of the type (a/S) appears in each term while one factor of each of 
the types ax and /3y disappears — this completely establishes our 
lemma. 

Ex. (i). Prove that Q'' a '" 6/ = , , , , ^ (ahY a^-"" 6 "-^ 

Ex. (ii). With the notation of the text prove that Q'' F. Q contains every 
term of type there written r ! times and that the number of different terms is 

, '~^, J — ". , — , . (Use induction.) 

(m — r)! («.-r) ! r! ^ ' 

28. Fundamental Theorem. Suppose now that 
P(ao, ai, ... a„) 
is an invariant of the binary form 

(tto, ai, ... a„][^i, x.^"" = a^" = ^^« = ..., 
then after the linear transformation 

^2 = ^2^1 + •J72^! 



26 THE ALGEBRA OF INVARIANTS [CH. II 

we have seen that a^ becomes aj and a^ becomes a, ; so that if 
the new form be 

we have Ar = aj^^^a/. 

By definition 

F(Ao, A„ ... Ar,) = (I1172 - hv^yFia,, a„ ... a„) ; 

accordingly if the A's are replaced by their symbolical expressions 
F becomes the sum of a number of terms, say XP . Q, wliere P 
contains only factors of the type a^, and Q only those of the 
type a,. As the degree in ^ and rj must be w we infer that there 
are just w factors in P and w in Q. 

If we operate on both sides with D,'^, P . Q becomes the 
sum of a number of terms each of which is the product of w 
factors of the type (ayS), and the result on the right-hand side is a 
numerical multiple of J^(ao«i ... ««)• 

Hence we have expressed F(ao, ai, ... an) in the symbolical 
form peculiar to invariants. 

The proof as given applies to invariants of one binary form ; 
it is the same, word for word, for any number of binary forms, for 
the left-hand side is still of equal degree in f and rj, and on the 
right-hand side we have the determinant {^r)) occurring to a power 
equal to this degree. Hence operating as above the required 
symbolical expression is obtained. 

29. The proof for covariants is of the same nature as that 
for invariants, although a little more care is required in the 
manipulation of the symbols ; after what has been said on in- 
variants we may confine our attention to covariants of a single 
form. 

Suppose that F{ao, a^, ... an, oci, x^ 

is a homogeneous covariant of order m of 

(tto, Oi, . . . ttn^iCi, X^y = Oa,^ = /8/ = etC. 

Then using the same notation as before 
F{Ao, Au ... An, X^, X^) = (^iV^-^-iViyPicio, ai. ••• ««, ^i, ^2). 



28-30] THE FUNDAMENTAL THEOREM 27 

If the A's are replaced by their symbolical expressions we get 

where P involves only factors of the type a^ and Q only those of 
the type a^. 

But on solution we have 

fl'72-t2'7l Cl»72-?2% 

Now for convenience we shall replace x^ by Wg s-nd a;^ by — Mi, 
so that 



X, = ,, "% , , Z, = - 



Wf 



(|l'72 - ^2'7l) ' (|l';2 - f2'7l) * 

Substituting these values and multiplying up by (^rj)^ we 
obtain the identity 

2 (- 1)"*= P . Qu^'^iu^'"^ = (^t; )«-+"» i^. 

We may write the left-hand side XP' . Q' where P' only 
contains factors with suffix f and moreover exactly (w + m) 
factors, and Q' contains {w + m) factors with suffix 77. 

Accordingly after operating with 11^+*" each term will involve 
{lu + m) determinantal factors, and the right-hand side will be 
a numerical multiple of F. Now of the {w-\- m) factors, there are 
w of the type (ayS) and m of the type (aw), for u must occur to 
degree m in the final as well as in the original expression. 

But {au) is — OLx ', hence replacing the it's by the xb throughout 
we have the symbolical expression for F. 



30. Since we have proved that all invariants and covariants 
of one or more binary forms can be completely represented by 
products of factors of the types (a/8) and ax, and further that 
every expression which can be so represented is an invariant 
or covariant — provided it possesses an actual significance, — it 
follows at once that all properties of invariants and covariants are 
implicitly contained in the symbolical representation and can be 
deduced therefrom. 



28 THE ALGEBRA OF INVARIANTS [CH. II 

31. Let us examine somewhat more closely the constitution 
of invariants of a single binary form 

/= (tto, tti, . . . an^xi, Xof = OL,'' = ^x" = 7*" etc. 

Suppose that an invariant / is an aggregate of products of 
factors (a/S) such that in every term there are w factors, then 
inasmuch as each symbol occurs n times in / we must have 

ni = 2w, 

where i is the number of different symbols. 

Now the weight of a,, is r by definition and its symbolical 
equivalent is ai^'^a/; hence the weight of any product of the 
as is the sum of the weights of the factors and is therefore equal 
to the total degree to which the letters ««, ySg, 72, ... occur in the 
symbolical equivalent. 

In the case of an invariant such as / each term in the 
symbolical expression when multiplied out is the product of w 
symbols with suffix 1 by w symbols with suffix 2 ; hence the 
weight is w. Further, the multiplying power of the determinant 
for I is also w. 

Consider next a covariant of degree i and order m. It is an 
aggregate of terms, each of which is the product of the same 
number (say p) of factors of the type (a/8), by the same number 
(say q) of factors of the type Ox. 

We deduce at once the relations 

q = m, 2p + q — ni, 

for each member of the latter equation represents the total degree 
of the covariant in the symbols a, /S, 7, 

Thus m = q = ni— 2p. 

32. The leading coefficient of the covariant is called a 
seminvariant — it is found at once from the covariant by putting 

cci = l, X2 = 

and therefore represented symbolically it is an aggregate 
of products of p factors of the type (a/3) by q factors of the 
type ai. 



31-33] THE FUNDAMENTAL THEOREM 29 

The weight of this seminvariant is accordingly p, and hence 
we infer that if w be the weight of a seminvai'iaut and i its degree 
the order of the corresponding covariant is ni— 'ho. 

Thus for example in connection with the cubic 

we have the covariant {a^ya^^x- 
The seminvariant is 

i.e. its weight is 2, as we should have inferred from the number of 
factors (olIS) in the covariant. 

The order of the covariant is 2 and here we have 

i =2, n — 2, 7n= 2, w = 2, 

so that m = ni — 2w. 

In like manner if the leading coefficient of a joint covariant 
of two quantics of orders Jij and n^ be of degrees i^, 4 in the 
respective coefficients and of total weight w in these coefficients 
conjointly, then the order of the covariant is 

riiii + 7?2t2 — 2w. 

The reader will readily establish this theorem and extend it 
to the case of any number of quantics by using the symbolical 
notation. On putting 771 = we get a relation connecting the 
degrees and weight of an invariant. 

33. Deduction of a covariant trova. its leading coeJBEicient. 

As we have seen, each term in the symbolical expression for 
the seminvariant must be the product of w factors of the type 
(a/9) by m factors of the type ai- 

Now suppose that in the seminvariant we replace ai by CLg, 
A by ^x, etc., and leave unaltered a., /S.,, etc., then (a/3) becomes 

(oiiTi + aoX^ A - (ySia?i + ^^x.^ tta = (a/3) on^ ; 

hence the seminvariant S is clearly changed into x^^ multiplied by 
the corresponding covariant — e.g. in the cubic, (a/3)"^ai/3i becomes 
xi^ X {oi^foix^x- We have thus a simple means of passing from 
the leading coefficient to the covariant. A similar result for 



30 THE ALGEBRA OF INVARIANTS [CH. II 

invariants may be obtained by taking the particular case m = ; 
here the leading coefficient is of course the invariant itself. 

Let there be an identical rational algebraic relation among 
a number of seminvariants S^, S2, ... Sr and let G^, G^, ... Gy be the 
corresponding covariants. 

If the relation be 
and Wp be the weight oi Sp, then the sum 

must be the same for every term — hence if we put the left-hand 
side of the relation into symbols and then change a^ into ax, 
/9i into ^x as above, we have 

or 2(7j'*'C/»...a/' = 0, 

i.e. the covariants are connected by the same relation as the 
seminvariants. 

34. Again when we replace Ui by Ox and leave a^ unaltered 
we replace the coefficient a^ by 



ax^-^OLr= 



_( n-r)l 9^ 
n! dx/ 



Hence except for a multiplier, which is a power of x, a 
covariant is the same function of 

/ 1 _§t 1 dY {n-ry. ay i ay 

as the corresponding seminvariant is of 

©0 , Ctj , £12 > • • • ^r > • • • ^n • 

Ex, (i). If in a seminvariant of weight w we replace ag by a^, /Sg by ^x, 
etc., and leave aj , ^^ . . . unaltered, then the result is the seminvariant multiplied 
by ^Tg". 

What is the corresponding transformation of the actual coefl&cients ? 

Ex. (ii). Find the result of replacing a^ by a* , og by a,, , ^i by /Sf , 
/Sj by ^T, , etc., in a seminvariant, and give the corresponding transformation 
of the coefl&cients. 



33-35] THE FUNDAMENTAL THEOREM 31 

Ex. (iii). Extend all the above results to the case of two or more binary 
forms. 

Ex. (iv). Prove that if in an invariant of a single binary form a^ be 

(n-r)\ d^f 
replaced by 7-^ ^—^ and so on, the result is the invariant multiplied by 

(n-r)\ d^f 
x^. State the result of replacing a^ by -^ j-^' ir-^ and extend the argument 

to any number of binary forms. 

35. Alternative proof of the Fundamental Theorem — 
The Aronhold Operator. 

We shall now give another proof of the theorem of § 28 in 
which the original argument of Clebsch will be followed. 

Let be a covariant of a form 

/=(ao, tti, ... an][a?i, x^"" 

which is homogeneons in both the coefficients and the variables, 
and of degree i in the former. 

If F = {A„A,,...An\X,,X^- 

be the transformed quantic, we have 

</)(J.o, ^1, ... ^„) = /A</> (tto, ai, ... a„) 

where /x. depends only on the transformation. 

Now if (&o. ^1, ••• ^n][«i, ^2)" 

be a second form which transforms into 

then (tto + >^^o, tti + X6i, ... an + ^h^x^, x^"^ 

transforms into 

(^0 + Xi?o, 4i + \5i, . . . ^« + X5„][Z„ X^f . 
Therefore 

</)(A + ^5o, ^i + X-Bi, ... ^„ + \5,,) 

= yu,^ (tto + X6o, tti + X6i, . . . a» + A.6„), 
hence expanding by Taylor's Theorem and equating coefficients 
of X we have 



32 THE ALGEBRA OF INVARIANTS [CH. II 

therefore the expression on the right is a joint covariant of the two 
forms ; in other words the property of invariance is not affected 
by an operator like 

. da) "" V ° 9^0 ^9^1 * * " " da J ' 

Hence, proceeding exactly as in § 14, we can construct a covariant 
of i different quantics which is linear in the coefficients of each, and 
which becomes a numerical multiple of (f> when we replace each of 
the i quantics hyf. 

The operator (&^) is called the Aronhold operator; its 

importance lies in the fact that it enables us to construct simul- 
taneous invariants or covariants of several binary forms of the 
same order when any invariants or covariants are known for a 
simple form. 

Thus, for example, since aoa-i—a^^ is an invariant of the quadratic 

(I/Q OC-y "7~ ^ (X\ w/j 00)^ "^ ttg U/2 

the expression aJ)^-{-aJ)Q— 2ai6i is a simultaneous invariant of the 
two quadratics 

(tto, ai, a^x^, x^^Y and (6o, h, h^x^, x^\ 

The construction of other illustrations will present no difficulty. 

36. It has been proved in § 14 that any covariant of degree i can be 
symbolically represented as a function </> of degree n in the coeflBcients of each 
of n different linear forms 

Oi, )3z> Jxy ••• '■> 

since ^ is a covariant of the original quantic it is unaltered by any linear 
transformation, except for a factor which depends only on the transformation, 
hence also it is a covariant of the i linear forms. 

By further use of the Aronhold operator we can now find a covariant of 
ni different linear forms 

a^W, a;,(2), ... ax(") 

)3:c(i), 0A ... /3x(») 
etc. 

linear in the coefl&cients of each form, and such that it becomes a numerical 
multiple of the original covariant when each of the symbols 

a(»), a(2), ... a(») 

is replaced by a, each of the symbols 

^(1), /3(2), ... /3(») 
by /3, and so on. 



35-37] THE FUNDAMENTAL THEOREM 33 

We need therefore only consider linear covariants of linear forms in the 
sequel ; we shall prove that every covariant of a system of linear forms is a 
rational integral function of invariants of the type (a/3) and covariants of the 
type ax- Once this is established the general theorem follows immediately. 

37. System of Linear Forms. First consider a single linear form 

and let (p be any invariant or covariant. 

If we use the linear transformation given by 

where b is any constant, then the new linear form is Xi and we have the 
equation 

where fi depends only on the transformation, i.e. only on a^, ag, b. 

Now let (f) = -\}rQai^ + ylria{'~^a2+ ...+'^r<i2' 

where the \|/'s do not depend on oj, ag but contain only x^, x^. 

Then *=%J/ + ^I'i^i'-M2+...+*r^2'' 

where ^^ is the same function of JSTj, Xg as x//-^ is of o^j, ^g > ^od further -4^ = 1, 
^2 = since the transformed form is X^. 

Hence *o = M^ = /^(>^oai'' + ^iai''"^a2+---+^ra2'") (I)- 

Now ^0 depends only on X^, JTg, therefore it is of the form 
C,Xi'»+(7iXi™-iX2+... + (7^Z2-, 

the C"s being numerical, hence equating coefficients of x{^ in the equation (I) 
we find 

C^a{^ = iLX 
where X does not depend on b. 

Consequently /n does not depend on b and therefore on making 6 = 
in (I) we find 

CoXi™ = ;x<^, 
for X^ is now zero. 

Hence /x is constant and (/> is a numerical multiple of Xj"*, i.e. of 

Thus a single linear form has no invariants and the only covariants are 
powers of the form itself. 

We shall now assume that a covariant of any number, less than n, of linear 
forms which is linear in the coefficients of each form can be expressed in 
terms of invariants of the type (a/3) and covariants of the type ax. 

Let <^ be a covariant of the same nature of n linear forms 
««» ^x) yxj 5*. ."J 

G. & Y. 3 



34 THE ALGEBRA OF INVARIANTS [CH. II 

let 01 be the result of putting ^=a in (^, (^2 ^^e result of putting y = a in (f)^, 
and 80 on, so that 0„_i is the covariant of the single form Og obtained by 
making 

a = ^ = y=... 

in (j). 



Now since is linear in /3 we have 



and in like manner 



'^'^V'hi^"'^)'''' 



etc. 



Consider (a, g- + o^ ^l) <^ = 0i 



as a differential equation for (f>, it being given that 0i does not contain 3. As 
a particular solution we have 

where *S' is the degree of (^j in a, i.e. S= 2 in our case, therefore 



where O^ ) =^i^ +^29^, 






and further yj/^ is linear in ^. 

Accordingly ^^1 = ^1^1+ P2&2 » where F^, P.^ do not contain ^ and 
aiPj+a^P^^O, 

• ■ „ ^ ^1' 

02 «i 

where xi does not contain /3 and is integral in a, y, 8, etc. 
Hence V'i = A^i+A^2 = («)3)xi 

and <^ = | (^ 9^)01 + («^);tl• 

But is a covariant, and ( ^ ^ j 0i is a covariant ; therefore {ai3) xi is a 

covariant ; again, (a^) is an invariant, therefore xi is ^ covariant. Moreover 
since xi(°^) i^ linear in a, ^, y, S, ..., xi is linear in y, S, ... ; therefore by 
hypothesis it can be expressed in tenns of (yS), y^, 8^, etc. 

Thus (f) = -(^^\(f)i together with an expression depending only on (a^), 
Qxy and factors of these types. 



37] THE FUNDAMENTAL THEOREM 35 

Then in like manner 

therefore * = 2^(^a-a) (^Da) ^^ 

together with terms of the required form. , 

Proceeding in this way we finally have 

*4(4)(ra^)('a^)-*. 

together with terms of the required form. 

But ^„ being a covariant of a^ is a numerical multiple of 

a." 

and therefore (^ 8^) (v ^) (4) * " ' '^" 

is a numerical multiple of a^^xyx •••• 

Thus we have expressed the covariant completely in terms of the two 
types of factors (a/3) and a^, that is, if the theorem is true for less than n 
forms it is true for n forms ; but it has been proved true for one form, hence 
it is true universally. 

Q. E. D. 



3—2 



CHAPTER III. 



POLARS AND TRANSVECTANTS. 



38. Two sets of variables 

Xi, ^2 > • • • ^n ) 

yi,y^., ••• Vn, 

there being the same number of variables in each set, are said to be 
cogredient, if, when one set is transformed by any linear trans- 
formation, the other set is transformed by the same transformation ; 
thus if a?i, a^j, ... x^ become Xj, X^, ... X^ where 

•^1 ^^ ^1,1-Ai + 'i,2 A2 T ... "I n, n-An 
X2 = ^2, 1 -^ 1 > ^2, 2 -^ 2 "I" • • • I ^2, w ^ n 



.(I), 



then 2/1, 2/2, 



•^n — 'n, 1 A 1 + In, 2 A 2 4" ... "T l'n,n-^n 

yn will become Fj, Fg, ... Fn where 

2/1 = ^1,1 -t 1 + ^1,2 ^2 + ••• H" n.n-'^n 
2/2 == ^2,1 ^1 + %2 -''^2 H" • • • + ^2,n ^ n 



.(11). 



Two sets of variables 

Xi, X2, ... ^n 
2/1.2/2, ... 2/n 

are said to be contragredient if, whenever the first set is transformed 
by the equations (I), then the second set of transformed variables 
Fi, Fg, ... F„ is given by the equations 

1^1 = ^1.12/1 + ^2,1^2+ ••• +ln,iyn 
-» 2 = ^1,22/1 + ^2,22/2 + • . . + l"n,2yn 



l^n= ^i.n2/i + ^2,ny2 + • • • + In.n^n, 



.(III). 



38-39] POLARS AND TRANSVECTANTS 37 

It is easy to see that if the set of variables x^, x^, ... x^ is 
contragredient to y-^, y^, ... yn\ then 3/1, 3/2, •.. 2/n is contragredient 
to iTi, x^, ... Xn. For example the symbols a and a; in a symbolical 
product are contragredient. 

39. From these definitions we deduce the following theorem : 

If x^, X.2, ... Xn; y-i, 2/2 > ••• Vn (^^'0 two contragredient sets of 
variables, then 

x^y-i + x^y^ + . . . + ^n1/n 

is unaltered by any linear transformation. 

Conversely, if 

^i2/i "t" ^2^/2 + . . . + ^n^n 

is unaltered by any linear transformation, the two sets of variables 
x-i, X2, ... Xn', 3/1, 2/2, ••• yn cc'^s contragredieut. 

Let the two sets of variables be contragredient, then if (I) 
and (III) be the equations of transformation, 

a-'i 3/1 +^23/2 "r ••• -^ ^nyn 
— ■Vi {U,\ -3^1 + ^1,2 ^2+ ••• +^l,7l-X',i) 

+ 3/2 (^2,1 ^\ + ^2,2 -3^2 + • • • + 4,n ^n) 
+ 

I Vn ('»i,i -^1 + 'n,2 -^2 '^" • •• "^ ''n,n ■^n) 
= ^1 (^1,1 3/1 + ^2,1 3/2 + ... + ln,xyn) 
+ ^2 {li,2 Vi + 4,2 3/2 + • • • + ^n,2 Vn) 
+ 

-\- 2Ln ('1, 71 3/1 + ^2,w 3/2 + ••• + ln,nyn) 

= Xi7i+X2F2+... + Z„F„. Q.ED. 

Conversely let 

«i3/i+ ^23/2+ ... +Xnyn 

be invariantive, then if «i, x^, ... Xn are transformed by equations (I) 

Xi Y^ + X2 Y.2+ ... + Xn Yn 
= x^yi + ^23/2 + . . . + x^yn 

= 3/1 (4,1 ^1 + 4,2 X^+ ... + 4,n -^n) 
+ 3/2 (4,1 X^ + 4,2 X2 4- . . . + 4,n Xn) 

+ 

+ 2^»(^n,l -X"! + ^^^ 2 -X'2 + . . . +^„_„X„). 



r^w 



6y9(>3 



38 THE ALGEBRA OF INVARIANTS [CH. Ill 

But this is an identity true for all values of Xi, X^, ... Xn\ 
hence the coefficients of Xj, X^, ... Xn on the two sides of the 
identity are equal ; i.e. 

Yl = ^1,1 2/1 + ^2,1 3/2 + • • • + ^n,l Vn 
Y2 = 11,2^1 + 12,2^2+ •-■+ ln,2yn 

^n — ^1, ra Vl + %m 2/2 + • • • + ^re,n Vn- 

This set of equations is the same as (III), and hence the two 
sets of variables are contragredient. Q. E. D. 

It may happen that a set of variables x^, x^, ... Xn is subject to 
a restricted group of linear transformations, and that 

Xiyi + ^2y2+ ••• +Xnyn 

is an invariant for all transformations which are allowed. The 
above proof still holds that y^, y^, ... 2/„ is contragredient to 
x^, x^, ... Xn. But yi,y-i, •-. yn is subject to a restricted group of 
transformations. 

Ex. (i). If the binary quantic 
become after any linear transformation 

(-^OJ -^15 -^2* ••• -^ni-^i, -A2)", 

then (tto, «!, ... ctnl^i, •^2)"=(^o> ^i> ••• ^n$J^n -^'2)" 



Hence 



■^OJ ( I j -^n (2]'^^' *■■' '^'" 



•*'! > "^l •''2> •••» •*'2 » 

are two contragredient sets of variables, subject to a restricted group of linear 
transformations. The linear equations connecting the original with the trans- 
formed coeflBcients of a binary form may be deduced from this fact (cf. § 16)*. 

* By means of the group here indicated binary invariants and covariants are 
brought under Lie's general theory of invariants of continuous groups. If we pass 
from the variables fj, ^^, ... tm to ^^^ variables fj', fj'* •■• fm' hy means of the 
transformations, 

fp'=/p(fl, fs' ■■• imJ <*l' «2» ■•• «r)» 
and these transformations form a group when the parameters a vary, then a 
function 



39-40] POLARS AND TRANSVECTANTS 39 

Ex. (ii). Two sets of variables both contragredient to a third set are 
cogredient with one another. 

Ex. (iii). If the determinant of transformation of a set of variables be /x, 
the determinant of transformation of the contragredient set is - . 

40. If F(ao, tti, a.2, ... a;^, x^) be any covariant of a binary 
quantic (uq, a^, a^, ... a^^x^, x^''^ and if y-i, y^ be a "pair of variables 
cogredient with x-^, x^, then 

y^dx^y^dx^ 

is unchanged by any linear transformation of the variables, except 
that it is multiplied by the same power of the determinant of 
transformation as that by which F is multiplied after trans- 
formation. 

For if 

F{Aa, A^, A„, ... Xi, X^) = fiF{ao, a^, a^, ... x^, x.^ 

where A^, A^, A^, ..., X^, X^ are the transformed coefficients and 
variables, and if z■^, z^ be any pair of variables cogredient with 
x-^, x^ which become Z^, Zo after transformation, then 

^(^0, ^1, ••• Z„ Z,) = fMF(ao, ai, ... z„ z,), 
fi being a power of the determinant of transformation. 

But Xi + Xyi, X2 + \y2, \ being any constant, are a pair or 
variables cogredient with x\, x^, hence 

F{Ao,Ai,...,X^ + \Y,,X. + XV.^ = fiF(aQ, ai,...,Xi + Xy^, x^ + Xy^). 

Now F being a rational integral algebraic function of all 
its variables, we may expand each side of the above equation in 
powers of X, ; since A. is arbitrary, the coefficients of the different 

is said to be an invariant of the group if 

^(fi'. fa'. •■•fm') = -P'(fl.f2.•••U• 
In our case the variables are (n + 3) in number, viz. a„, aj, ... a„, Xj, acj, and the 
transforming equations are all linear. The parameters are the coefficients in the 
original transformation of .Tj, x^, and if they be so chosen that Iii7.2~^2'?i = l- ^^^^ 
an invariant or covariant of the binary form is an invariant of the above group 
in (n+3) variables. The reader will find it interesting to write down the actual 
transformations for the a's and thence to verify that they form a group. Cf. Lie, 
Vorlenurifien ilber contiimierliche Gruppen, p. 718 etc. 



40 THE ALGEBRA OF INVARIANTS [CH. Ill 

powers of \ must be the same on the two sides of the equation. 
Hence, using Taylor's theorem : 

F{Ao, ^1, ..., Zi, X2) = fMF(ao, tti, ..., Xi,x^) 



( ^i^Y + ■^^ov^ ) -^(-^O) -4i, ..., Xi, X2) 



from which we see that f 3/1 x-;- + 3/2 ^— ) -^ is invariantive, if F is 

\ OOl^i OtX/2 1 

a covariant. 

41. The definition of covariants may now be extended thus : 
Any function of the coefficients of a single quantic, or of a 
simultaneous system of quantics, and of sets of variables, — all sets 
being cogredient with the variables of the quantics, — which is 
such that, when any linear transformation is made and the 
original coefficients and variables replaced by the transformed 
coefficients and variables, it is unaltered except for a factor de- 
pending on the coefficients of transformation. 

Let a^i, 5^2 ; 2/1, 2/2 be two cogredient sets of variables, then 

{x^y^-x,,y:) = {xy) 

is unaltered, except for a factor — which is the modulus itself — by 
any transformation. 

Hence, if it be necessary to consider covariants with sets of 
cogredient variables, we may replace all the sets of variables but 
one by the coefficients of linear forms added to the system. For 
we may regard {xy) as a linear form — since it becomes (ZF) 
by transformation — and hence replace the variables y^, —y^ by its 
coefficients. 

Hence covariants having more than one set of variables may 
be represented symbolically as a sum of products of symbolical 
factors of the types (a/S), ««, Uy, {xy), — 

42. Convolution*. The word convolution is used as a 
name for the process of obtaining from a given symbolical 
product (representing a covariant) another symbolical product 

* German Faltung. 



40-43] POLARS AND TRANSVECTANTS 41 

(representing another covariant) by removing two of its factors 
of the form cLx^x and replacing them by a single factor of the form 
(ayS). Any symbolical product P' obtained from the product P by 
means of this process either once or several times repeated, is 
said to be obtained from P by convolution. Thus the covariant 
{ahf{ac)aJ)^Cx of the quartic is obtained by convolution from 
the covariant {ahfa^b^Cx*; the factor a^Cx being replaced by {ac). 
It may also be obtained by convolution from {ah){ac)axhxCx', 
also from {ah)axhxCx*, or from axhxCx- 

This process, it should be noticed, is purely a symbolical 
process, and has no analogue in the non-symbolical treatment of 
modern algebra. 

43. Polars. The operator 

has already been introduced. 
In § 15 we defined the form 

^'(^a^)> (^-) 

to be the rth polar of P ; P being a binary form of order n. 

Again in § 40 of this chapter it was proved that if P is a 
covariant and the variables 3/1, y^ are cogredient with x-^, x^, then 
the form (IV) is also a covariant. 

For the purpose of calculating polars, we may use a theorem 
identical with that of Leibnitz for ordinary differentiation. 

Thus if the ?'th polar of a^h^ 

be required, it may be obtained by operating on this expression 
with 

(m + »)! 

where Pj = y^ ^ — + 2/2 ^ , but operates only on ax'* ; and Pg is 

the same expression but operates only on bx^. 

Ex. (i). Find the .second polar of (ab)- ajbj. 
Ex. (ii). Find the rth polar of a^'^h^^cjj'. 



42 



THE ALGEBRA OF INVARIANTS 



[CH. Ill 



44. In view of the fact that covariants will generally be 
given in terms of symbolical letters which refer to the original 
quantic or quantics, and that in this case the factors a^ are not all 
the same, we shall consider the general case in which these factors 
are all different. Results, proved for this case, may be obtained 
for any other by simply equating two or more letters. 

Consider now the form 

where P is a product of symbolical factors not containing x-^, x^. 
The first polar is 

" fl V ' ' X 





+ 


Is 

n 




•••««-!, "nj 



the term in the bracket being simply an abbreviation for 

We notice in this expression that : 

(i) The difference between two terms 



L «r 



-F'^.oi,^ 



y " - 



] 



L «r a« J 
where X is an expression obtained from F by convolution. 

(ii) The difference between the whole polar and one of its 
terms 



F''.ai^-^ 






i^*".a. 






POLARS AND TRANSVECTANTS 



43 



44-45] 

where X is a sum of terms each obtained from F by convolution, 
each such term being possibly multiplied by a constant. 

Similarly the rth polar of F^^ is 

fli otg . • . Or 






F. 



a, cCi 



(V), 



where the summation extends to all possible sets of r factors 
taken from a.-^ a-i ... a« . To see the truth of this suppose it true 
for the rth polar, then the r + 1th polar may be obtained from 

the 7'th by operating with — • (y^)- ^o the left this gives 



n — rX' dxj 

■p^n-r-i p^r+i^ Consider one term on the right, that expressed 
above ; by polarizing we obtain 

«! tta ... <Xr di ' 



r\{n-r -1)1 ^^ 



F^. 



«! Oa 



Clr °li 



Each term of (V) gives rise to a similar expression. Now any 
particular term of the r + lth polar, arises r + 1 times, once 
from each of r + 1 terms of the rth polar, thus 

" tti a^ ...Or Or+i " 
J/T n y y » » 

arises from each of those terms of the rth polar, obtained by 
omitting one of the factors in the numerator and the corresponding 
factor of the denominator of this expression. Hence 

ai fla • • • O^r+i ^ 
, y y _y 

aj^Og^ ... a,.+i^ 

Now it has been seen that this is the correct form for the first 
polar ; hence it is the correct form for the second, and so on ; the 
formula is therefore true in general. 

45. The number of different terms 

^ «! "2 • • • Or ^ 

FJ^ ' " -' 



pn-r-. p ,+, ^ (r+l)!(»-r-l)! ^ 

" n ! 



F^ 



18 



IS 



The coefficient of each term as it appears in the polar 

F^^'-^F/ 
1 



0' 



44 THE ALGEBRA OF INVARIANTS [CH. Ill 

hence the sum of the coefficients of all the terms of a polar is 
unity. 

This remains true if some of the letters a become equal, for no 
term of the polar can vanish. 

46. Two terms of the rth polar are said to be adjacent, when 
thev differ only in that one has a factor of the form a^ aj, while in 
the other this factor is replaced by a^ a* • 

We shall now prove that : 

(i) The difference between any ttvo terms of the rth polar of 
F^ is equal to {xy) X, where X is a sum of terms each of which is 
a term of the r — 1th polar of an expression obtained from F^^ by 
convolution. 

(ii) The difference between the rth polar of F^ and any one 
of its terms is equal to (xy)X, where X is a sum of terms each of 
which is a term (multiplied by some constant) of the r — 1th polar 
of an expression obtained from, F^ by convolution. 

The difference between any two adjacent terms 

%\ F, - ccha^^ F, = (xy) (a^afc) F^ = (xy) X 

" ' , r i?!," ~ . 

where X is a term of the ?•— 1th polar of (a^ajk) ^ ~ , *'-6- a 

term of the r— 1th polar of an expression obtained from F^ by 
convolution. 

Now any term of the rth polar of F^'' may be obtained from 
any other by means of a finite number of interchanges of letters 
such as ail, aj^. Hence between any two terms Ti, T^ of the rth 
polar a series of terms Tj^,, Tj^j, ... T^^i may be placed such that 
each term of the series 

-^1) -^1, i> -^1,2) ••• -'i, I) ■'a 

is adjacent to that on either side of it. Hence the difference 
between two terms 

T,-T,= {T,-T,,,) + (T,,,-T,,.} + (T,,,-T,,,)+ ... +(T,,i-T,) 

= (xy)X 

where X is a sum of terms each of which jis a term of the r — 1th 
polar of an expression obtained fiom F^^ by convolution. 



45-47] 



POLARS AND TRANSVECTANTS 



45 



Again, the difference between the complete polar and any 
single term T 

Hr) 

= 2 

i = l 



T, 



-T 



Ti representing the general term in the polar 

'T,— f 



Hr) 
S 



and hence the theorem (ii) follows at once by (i). 

It should be observed that if two or more factors are now made 
identically equal, the above proof is not affected ; we may for 
the sake of argument suppose them all different, and make them 
equal in the final result. The only effect of the equality of 
factors is that some of the terms obtained by convolution from 
Fx^ will vanish. The propositions are true, then, as stated, for 
any symbolical product ; and consequently for any covariant form. 



47. Let T be any term of the rth polar of F^^, then by 
proposition (ii) of the last paragraph 



where ^^_i is a term of the 5 — 1th polar of a form obtained from 
Fx^ by convolution and \y_i is nurnerical. Let the r— 1th polar, 
of which (jjr-i is a term, be -v|r,._i. Then applying (ii) again we 
have 

(f)r-i — ^{rr-i = i^y) 2 Xr-2 4>r-2 



where ^r-2 is a term of the r — 2th polar of a form obtained by 
convolution from F^^. 

Proceeding thus we see that 

where ■yjr^. is the Jcth. polar of a form obtained from F^^ by con- 
volution ; and A^ is numerical. 



46 THE ALGEBRA OF INVARIANTS [CH. Ill 

The terms of this series, the existence of which has just been 
demonstrated, will be accurately determined later. The series 
is known as Gordan's series. 

48. Transvectants*. If aa;"*, 63:'' be any two binary quantics, 

the form 

{aby a/"-*- 6/-*- 

is called their rth transvectant, or their transvectant of index r. 
The symbol 

(/ <f>r 

is used to denote the rth transvectant of two forms/, <f). 

Thus 

(a/'^, 6/)'- = {aby a^-^ b^\ 

The definition of a transvectant just given is symbolical : the 
process of forming a transvectant is however not a purely symbolical 
one, — like that of convolution. In order that we may be able to 
obtain transvectants of any two forms, we make use of the diffe- 
rential operator, introduced in Chapter I.: 

n = -^^ ^. 

Thus 

fn, t M ! 

" {m — ry. {n — ry. ' ^ 

Hence if f{x\ (f> {x) be any two forms of orders m, n re- 
spectively, then the rth transvectant of / and (f> may be obtained 

by operating with ^ 1 -^ — r- ' Cl^ on f{x).<f>{y) and after 

operation replacing y by x. 



Thus 

{m — ry. (w — r)! 



{f{x), <!> {X)y = ^^^^- . ^^^- [nrf{x) . (/> (2/)],., . 



Ex. For the cubic the first and second transvectants of the Hessian with 
the cubic itself are 

{{ahf a^b^, c^sy=^ {ahf {ac) Kc^^-^\ {abf {he) a^c,\ 

{{ahf ax hx , c^^f = {ahf {ac) {he) Cx ; 

as may be seen by using the differential operator. 

* German Vberschiebung. 



47-49] POLARS AND TRANSVECTANTS 47 

It is useful in calculating transvectants to notice that, just as 
in the case of polars, the sum of the coefficients of the various 
terms of a transvectant is unity. 

49. For the purpose of calculating transvectants the following 
method is extremely useful. 

Consider the rth transvectant of two forms aj^, h^\ it is 
{ahy a^"*-*" h^-^. 

It may be obtained by the following rule : 

Polarize a^^ r times with respect to y, we obtain a^^'"^ aj \ 
then replace y-^ by h^, y^ by — 6j and multiply by ho^~^, the result 
is {ahy »«:"'"'■ &»''"'■ which is the rth transvectant of a^ and h^. 

We proceed to illustrate the method : 

(i) Consider the second transvectant of the Hessian of a 
quartic with the quartic itself, 

The second polar of (ahy a^b^^ is 

^ (aby ay^bj^ + | (aby ajy^ayby + ^ {aby aj^by^ 
Hence the transvectant required 
= 1 {aby (acy b^'cj" + 1 (aby (ac) (be) a^b^c^^ + J {aby {bey a^d 
= \ {aby {acy bj'c^' + | {aby {ac) {be) a^b^c^^ 
since a and b are equivalent symbols. 

(ii) The third transvectant of these two forms is 
{aby{acy{bc)b^c^. 

(iii) To obtain the second transvectant of the Hessian of the 
quartic with itself: i.e. 

{{abya,'b,\{edye^'d^J. 

Let us write {cdy c^di = hx*, where the symbolical letter h 
refers to the coefficients of the Hessian considered as a separate 
binary form. Then as in (i) 

{{abya^'b^Mi^J 

= J {aby {ahy bJ'K^ + 1 {aby {ah) {bh) aJ)^K\ 
To obtain the first term we polarize h^* twice, replace y^ by — a^ 



48 THE ALGEBRA OF INVARIANTS [CH. Ill 

and 2/2 by + aj and then multiply the result by ^ (aby bx'- Thus 
the second polar of h^* 

= Hcdy Cy'dJ' + 1 (cdy CydyC^d^ + J (cdf C^Hy\ 

Hence 

\{aby{aKfbx'K' 

= jg {ahf {acf (cdy b^H^ + \ (aby (ac) (ad) (cdy c^dj)^ 

^^^(aby(cdy(adybx'c^ 

To obtain the second term we polarize h^* once with respect 
to y and once with respect to z, and then replace y hy a and 
z by b. 

The polar is 

^ (cdy CyCzdi + \ (cdy CydzC^d^ + \ (cdy CgdyCxd^ + ^ (cdy c^dyd^. 

Hence 

\ (aby (ah) (bh) a^b^hj" = ^ (aby (cdy (ac) (be) ajy^d^^ 

+ I (aby (cdy (ac) (bd) aj^c^dx + f (aby (cdy (ad) (be) a^bxC^d^ 

+ ^ (aby (cdy (ad) (bd) a^b^d. 

Hence remembering that all four letters are equivalent, we 
obtain 

((abya^^bx\(cdycHry 

= \ (aby (acy (cdy b^^d^ + 1 (aby (cdy (ac) (ad) c^dM 
+ f (aby (cdy (ac) (bd) aJ)xCxdx. 
(iv) Calculate 

(v) Transvectants of the following form are of frequent occurrence : 

= 2 Uy \m/ ^^^a ^j^ n-Xj^m-^ p-r 

The result may be obtained at once by polarization, 
(vi) From this may be deduced the value of 

Let Cj5Pc?,9=Aa,P + «. 



49-50] 

Then from (v), 



POLARS AND TRANSVECTANTS 



49 



T= 2 

<T-Vr=r 



CKI) 



{aKf {hhy a/- " h^"^-^ h^P+^-''. 



But polarizing Ax^^* f times with respect to y and r times with respect 
to z we have 



^ ^ X!.!(^-X-.)! ,!z.!(g-^-.r)! , 



cr\ t! (^ + g'-o- — r)! 
Hence replacing the y's by a's and the /s by 6's 



P! 



r = 



2 ?•! (n-X-^)! (m-i/-tzr)! (;?-X-i/)! (y-/x-ar)! 



{'m + n — r)\' {p + q — r)\ 
(vii) If /=ai a2^...a^ , 0=^, ^2,... /3„ , 



(/, <^r= 



1 sf (ai/3i) (02/32) -K^r) ^ ,1 



where the 2 extends to all possible arrangements of the letters aj, 02, ...a„; 
^i,)3„..A. §44. 

50. Two important theorems relating to the difference between 
terms of a transvectant, must now be proved. They are exactly 
analogous to those already obtained for polars in § 46. 

(i) The difference between any two terms of a transvectant is 
equal to a sum of terms each of which is a term of a lower trans- 
vectant of forms obtained by convolution from the original forms. 

(ii) The difference between the whole transvectant and any one of 
its terms is equal to a sum of terms each of which is a term of a lower 
transvectant of forms obtained by convolution from the original forms. 

Here, as in the case of polars, we introduce the idea of adjacent 
terms. Two terms of a transvectant are said to be adjacent when 
they differ merely in the arrangement of the letters in a pair of 
symbolical factors. Two terms can be adjacent in any one of the 
following ways : 

(i) P (ai^j) (a^^k) and P (a,/8,) (a^^j), 
(ii) P (ai^j) tth^ and P (ah^j) \, 
(iii) P (ai^j) ^[ and P (a^/S,) y8-^, 

G. & Y. 4 



50 THE ALGEBRA OF INVARIANTS [CH. Ill 

where the letters «!, otg, ... belong to the first of the two forms in 
the transvectant, while ^i, ^2,--- belong to the second form. 

The difference between two adjacent terms is in the three 
cases seen to be 

(i) P(«,ar,)(/3,-/3,), 

(ii) P{aia„)^j^, 

(iii) P(^,^j)ai^. 

To fix ideas we shall suppose that the transvectant we are 
considering is the rth transvectant of 

and (f) = B.^^^^^^...^n^, 

where A and B are products of symbolical factors of the type (yB), 
and all the factors of the type y^ are different. 

Then 



(/.<^r= ^ 



C) 



where the S extends to all possible arrangements of the letters 
tti, Oa-.-o^; /3i,^2-..^n,— § 49 (vii). 

The truth of this statement may also be seen by operating 
with Xl*" onf{x) (y) ; and remembering that if we write 

ai = a2= ...=a^ = a, /3i =^2= .•• =yS„ = /8, 

then (/ (f>y = (a^y a,"^-^/3^''-^. 

The difference between two adjacent terms of the above trans- 
vectant is a term in which at least one factor of the type (ayS) is 
replaced by a factor of the type (oca'), or else of the type {^fi'). 
There are then not more than (^ — 1) factors of the type (ay9). 

Hence the difference between two adjacent terms is a term of 
a transvectant of index less than r, of forms obtained by convolu- 
tion from / and <j). 

Thus, for example, 

(abf {acf h^ c^, (aby (ac) (be) aja^c^ 

are adjacent terms of the transvectant 

{{abya^^bi,c^y, 
and their difference 

{aby(ac)b^c^^ 



50-51] POLARS AND TRANSVECTANTS 51 

is a term of 

{{abfaj)^, Co;*). 

Now between any two terms Ti, T^ we may place a series of 
terms 

-'i, 1> -'i, 2» ••• -^ 1, i 

such that any term in the series 

1> -*^1, 1> -^ 1, 2> ••• -^ 1, 1> -^2 

is adjacent to that on either side of it. 

For we may obtain T^ from Ty by a finite number of inter- 
changes of pairs of letters, a pair being composed either of two as 
or else of two yS's ; since each letter occurs only once — in our 
argument, — the terms which differ by the interchange of a single 
pair are adjacent. Hence the difference between any two terms 

i.e. a sum of terms each of which is a term of a transvectant of 
lower index of forms obtained from the original forms by convolu- 
tion. This is the first theorem. 



Again if T be any term of the transvectant, then 



(/, (jiy-T= — !— - tr - T 



r) \r. 



2 {T - T) 



[since the number of terms T' is, r\{ j ( j J 

and this is equal to a linear function of terms of transvectants of 
lower index of forms obtained by convolution from y and <j>. 

51, This theorem may be extended by applying it again to 
each of the terms of transvectants of lower index on the right- 
hand side. This may be done repeatedly. Each time the process 
is applied the terms of transvectants on the right are replaced by 
the transvectants themselves, and linear functions of terms of 
transvectants of lower index. After not more than r applications 

4—2 



52 THE ALGEBRA OF INVARIANTS [CH. Ill 

of the process, we have on the right a linear function of trans- 
vectants of forms obtained from f, <f> by convolution, whose index 
is less than r, and of terms of transvectants of zero index. 

But a trans vectant of zero index of two forms is simply the 
product of the forms ; a term of such a transvectant is merely the 
same product, and therefore the transvectant itself 

We obtain then the following important theorem : The differ- 
ence between any transvectant and one of its terms is a linear 
function of transvectants of lower index of forms obtained from the 
original forms by convolution. 

Ex. (i). {abY(bcya^W 

= {iabf a,^h,\ c,*)2 - % {ahY (be) a,cj^ - J {ahf (ac) b^c^^ 
= ((a6)2 a^^bj', c^^Y - (Sflbf a^b^, c^*) + § {abf . c^\ 

Ex. (ii). Prove that 

{{ab)a,^bj^,{cd)cj^d,^f 

= ^iab) (cd) {{boy {adf+9 (bcf {adf {ac) (bd) 

+9 (6c) (ad) {acf {bd)^+{acf {bd)^} 

= J {(&«)* («^* - («c)* {bdf} - \ {abY {cdf {{bcf {adf - {acf {bdf} 

= i {Q>cf (ad)* - {acY {bdf) - \ {abf {cdf {{be) {ad) + {ac) {bd)} 

= J {{be)* {ad)* - {ac)* {bd)*} - 1 ((a6)3 a,b, , {cdf c^d^f. 

Ex. (iii). If /be a cubic and H'\\& Hessian, prove that {H,ff=0. 

Ex. (iv). If/ be a quartic and ZTits Hessian, prove that {ff,f)^ = 0. 

Ex. (v). If bx* be the Hessian of a^*, and {ac)* = = {ad)*, then 

{{ab)a^^bxMcd)ex^dx^f=0. 

Ex. (vi). The second transvectant of a quantic / of order n and its 
Hessian II is 

where i={f, /)*. 

For 

V 2 ) 

= (a*)2(ac)2 a,'»-46.»-2o.»-2- gj^^ t/ 
on using (III) § 22. 

Finally the result is obtained by the help of the last identity of § 22. 



CHAPTER IV. 

GORDAN'S SERIES. 

52. Gordan's series. In § 47 it was proved that any term 
T of the 7wth polar of a symbolical product F can be expressed in 
the form X{xyyFi; where Fi is the (m — i)th polar of a sum of 
symbolical products — possibly multiplied by positive or negative 
constants — each of which is obtained from F by convolution. The 
product a^hy^ is a term of the with polar of ax"})^, hence 

a,«6,- = 2(a;2/yi^<V-* (I); 

where the suffix indicates that F'^^^^-i is the (m — i)th polar 
of F^K 

This is an identity; it must therefore be true when y = x*\ 
hence 

for when y — x, {xy) = 0. Now if in a function of x polarized with 
respect to y, y is replaced by x, the result is the same as the 
original expression. Thus the rzth polar of a,^'^ is a^ay^, which 
becomes a^'^'^ when y is replaced by x. Hence 

a^H,-^ = \F<\m\^^=F' (II). 

Thus the first term of the series (I) is obtained, the other terms 
may be obtained in a similar manner. Operate on (I) with IP'. 
Consider first the effect of this operation on the term 

{xy)'F^\,.-i. 
* This of course means — = — . 



54 THE ALGEBRA OF INVARIANTS [CH. IV 

By comparing the degrees in x on the two sides of (I) it may be 
seen that the degree of F^^^^-i in ^ is n — i. Hence we may 
write 

and F^^ym-i = a^^-'a/'-\ 

Now 

= i(m+n-i + l) {xyf-^(i^-^(iy^-\ 
a result obtained by ordinary differentiation as in §§ 26, 27. 
Hence by repeated operation of VL we obtain 

\ y' ^ y {^-J)\ {m + n-i-j Jriyr ^' ^ v ' 

when i >j; but when i <j 

nJ {xyfaj'-^oiy'^-^ = 0. 

As regards the right-hand side of the equation (I) we know, § 27, 
Ex. (i), that 

nJaJ'by'^ = ; ~, . 7 ^, {ahya^^-iby^-i. 

(n-jy. (m-jy. ^ 

The result of this operation is, then, 

in-jy.'im-jyr^^'''' ^' 

= 2 y. — TT^, -. — -. — =-r-, {xyf •? J^'*V-<. 

i=i (t -;)!(w + w - 1 -J + 1)! ^ ^^ ^ 

In this put y = x, and we obtain 

(w-^)! (m-;)!^ * * (m + w-2j + l)! 

and hence 

F(3) = ^J^^3/ {abya^^'-^hx'^ 

in 






52-53] gordan's series 56 

Substituting this value of ^'■^"' in (I) we have Gordan's series : — 



^n\ /m 
i=o /m + n — i 






/m + n — i + 1\ 



53. This series may be put into other, rather more general, 
forms. 

Multiply by {aby, write « + ^ for n, and m + r for m, then 



«-.--.(M^)«<..-.«;. 

Operate with ix-^], and we obtain 

^n — r\ fra — r — k 
i 



fn — r\ fi 






Operate with [y ^] on (IV), then 



m—r 

'y 



^n — 1 — k\ fm — r 



= 2 



+„-2;-.>i\ ("^>'('"" *«-);:-— - (^i)- 



In (V) replace y^ by Cg, 3/2 by — Ci, and multiply by c^P"^'^^'^^, 
we obtain — see § 49 — 

'n — r\ fm — r— k 



7)C 



= X{-iy " " ' \ — V-^ ((aa;^ bary^'-, c^^)"»-*-^-*...(vii). 



56 THE ALGEBRA OF INVARIANTS [CH. IV 

The equation (VI) gives a similar result when the weight of 
the covariants under consideration is greater than m the order of 
fea,*": viz. — 

(aft)*- (6c)'™-^ (ac)*aa;''-^-*Ca;2'-"'+'-* 

/n — r — k\ /m — r\ 

= S ( - 1)^' L + L2r-/+l( ^^^»''^' ^*">'^' c.^)-i-+* . . . (VIII). 
\ i J 

54. Now if in (VII) a and c, n and p, r and m — r — k are 
interchanged, the left-hand side of the relation is unaltered, except 
for a factor ( — 1)"^~*. Hence 



fn — r\ .Vfi — r — k\ 



fm + n — zr -i+\\ 

'p + r-'tk — m\ fr\ 



= 2 ^ ^ ^ . :, ^-^-frv ((^*"'. 0^^)^+"^'--*, a*")''"*- 

'^ + 2r-l-2« — w — t + l\ 



On the supposition that 

f=h^^, <i> = aar, '^^cJP, a, = 0, a^ = m-r-k, as = r, 
this relation is the same as 

% A ! l\ll^((f A,)<H+i y}rY: + <h-i , 

= ( - 1)"' S ^^ \ .\ (( /, ^|r)-.+», <f,)».+«3-* . . . (IX). 

^ ^ /m+p - 202-1 + ly^-^' ^^ ' ^^ ^ ^ 

Again, if in (VIII) we interchange a and c, n and p, r and m — r 
respectively, the left-hand side of the relation is unaltered except 
for a factor (— 1)"»"^. 

Hence we obtain a relation of the same form as (IX), but 
in which 

ai = k, 0Lz = m — r, ct^ = r. 



53-55] gordan's series 57 

Thus the identity (IX) is true in two cases. In the first 
ai = : as cannot exceed either m or n, for we use in (VII) the 
factor {aby : 0^+ cis — m — r and therefore cannot exceed m : and 
finally a^ cannot exceed p. 

With these restrictions Oa, a.^ may take any positive integral 
values. 

In the second case 

tta + as = ini- 

ttj + «! cannot exceed n, and a^ + aj cannot exceed p ; subject 
to these restrictions a^, a^, a^ may take any positive integral 
values. 

The series (IX) is one of great importance for calculating 
transvectants. It is usually quoted in the form 

/ f <f> f \ 
{ m n p ) , 

\ «! 02 Ots / 



«•> + Ofs > W, tts + fli > n, «! + tta > p, 



where 

and either 

(i) fli = 

or 

(ii) aa + Qs = wi. 

55. In order to illustrate the use of the series, we shall now 
calculate some transvectants which are covariants of the sexticf*. 
Let us write 

H = {f,f)\ i = (f,fy, A=(f,fy, t = {f,H). 
Then to calculate the transvectant (H, fy we use the 
series 

/ / / / \ 
(666), 
\ 3 2 / 

(7) (7) 

* other examples will be found in Chapter V. 



68 THE ALGEBRA OF INVARIANTS [CH. IV 

that is 

{H,fr+^{i,f)={i,f), 

or {H,fr = \{i,f). 

The object usually in view when calculating transvectants, is 
to express them in terms of other transvectants of lower index. 

There is no difficulty in choosing the series which will give the 
desired result, in general we select it so that the transvectant to 
be calculated appears as the first term on one side of the identity, 
while all terms on the other side are of lower index. 

The transvectant (H,/)* is given by 
/ / / 



/ / / / \ 
(666) 
\ 4 2 / 



((/ /^^ fy-' = ^4P4 ((/ n*^'' /)'-'> 



hence 



CT) {'?) 



iH,fy + ^-^{i,fy+lA.f=(ify + lA.f, 



or {H,fy = ^(ify + ^A.f. 

To calculate (H, fy we cannot write ai = 0, for then the 
condition a^ + aai^m would be violated. 

We must then make ttg + ctj = m = 6 and use the series 
/ / / 



// / / \ 
(666) 
\ 1 4 2/ 



hence 



i ) K i I 

(ff,/)'=-y*'(i,/r. 

* This form vanishes identically — see Ex. (ii). 



55-57] gordan's series 59 

To find (t, fY we use 

// H f. 
16 8 6) 
\ 2 1 / 

whence using the result, § 51, Ex. (vi) 
we obtain 

Ex. (i). Prove that for the sextic 

(zr,/)«=^(t,/)*. 

Ex. (ii). By means of the series 

/ / / 

6 6 6 

13 3 
prove that (i, f)^ = 0. 

Ex. (iii). Prove that 

iif,^r',ff=-^QU^r,n 

{H, ^•)4=2^^•,^•)2 + ^^.^•. 

56. The series we have been illustrating does not give all the 
relations between the covariants of/, ^, -^ which are of weight 

«! + «2 + "s- 

For example, we cannot by means of this series reduce the 
form ((/, fy, fY which is reducible whatever be the order of/, 
§ 51, Ex. (vi). 

57. The series (III) may be inverted with the following result 

n («) 

(a,™, 6=.»)V = S(- l)*^)£;^(aJ)'a,"'-'6„»-«(^2/)' (X). 

i I m "T '^^ 

\ i J 
(Gordan, Invarianten-theorie, § 7, pp. 89, 90.) 



60 THE ALGEBRA OF INVARIANTS [CH. IV 

To prove this, we first establish the existence of an expansion 
of the form indicated ; and secondly we find the coefficients. To 
prove the possibility of such an expansion, we observe that 

= l\ia^^-%''-^ [aj)y - {ah) {xy)Y 
= S/Uj {aby {xyy a^-i hy^'-i. 

To determine the coefficients /u,, we operate on the relation 
just proved with ft ; the left-hand side of the relation vanishes, 
and hence by equating the coefficients on the right to zero we 
obtain 

yLtii (m + w — *■ + 1) = - /ii_i (m — I + 1) {n — i-\- 1). 

Replacing y by a?, we see that /io= 1, hence in general 
f^i = (-) 7 



m + ri 



Other series may be deduced from (X), by the same methods 
as were used in § 53. 

Thus 

f7n — k\ fn — k 



I / \ I 



(aoT, h^fyn-u = 2 {-y \^/^_2M («^)''*-*«^""'"* V"'-* (^)^ 



((«*"*, hx^^f, cjy 

'tti — k\ (r 



{ 



= S ^ — " ' 'V. (aby+''{bcy-' a^"^«-*&a;"-'-*Ca;^-'-+^' . . .(XI), 
/m + n-2kY ^ ^ ^ ^ ^ 

when r if-n — k. 

fm + n- 2k — 7'\ hi — k\ 

(K"*. &*")*, cx^r = S -^ r^-r 1 , ' 

X (a6)»+* (6c)"-*-* (ac )*+»-" a^m+n-2*-r-tc/-r+f (XII). 

58. It has already been pointed out, § 41, that the system 
of concomitants belonging to a binary form with two or more 
cogredient sets of variables, is the same as that obtained when 



57-60] gordan's series 61 

certain binary forms with only one set of variables are taken for 
simultaneous ground-forms. Gordan's series (III) shews us which 
the ground-forms must be ; thus for the form 

we consider the system 

ax^hx, {ah) ax, {xy). 

Each member of this system is unaltered by linear transforma- 
tion, and ax^hy can be expressed in terms of the forms given. 

59. Ex. (i). Any symbolical product having two cogredient sets of 
variables x, y, can be expressed in the form 

where P is a symbolical product containing only one set of variables x. 

If we put 

Xy^l, .^2 = 0, yi = 0, ^2 = 1. 

then Pym-i 

becomes the coefficient oi x-^~'^'^^x^~^ in P. 

Hence we may express any rational integral function of the coefficients of 
a binary form linearly in terms of the coefficients of its covariants. (Elliott, 
Proc. London Math. Soc. vol. xxxil. p. 213.) 

Ex. (ii). Express the product ao'^i*2 of the coefficients of the cubic 

(<^0> '^IJ '^2) ^X-*^!} •^2/ 

linearly in terms of coefficients of its covariants. 

60. If a symbolical product representing a covariant of a single 
binary form contain a factor (aby^~^, it may be expressed in terms 
of products each containing a factor (ab)^. 

Let the order of the binary form be n ; and let P be the 
covariant in question. Then since P contains the factor (a6)^^~^ 
it is evidently a term of a transvectant 

{(abf^-'ax''+'^^bx''+'-^\ (f))", 

where (f> is some other covariant. 

Hence by § 51 

P = ((a6)'^-iaa,"+i-^6a,"+i-=^\ (f>y 



+ tC {{aby^-'ax''^'-'^ 6^"+^-^, </>)"' (XIII), 

where C is a constant, and ^ denotes any function obtained from 
<fi by convolution. 



62 THE ALGEBRA OF INVARIANTS [CH. IV 

Now 

for h and a are equivalent symbols. Hence all those transvectants 
in (VIII) in which no convolution has taken place in the first 
form are zero. But every form obtained by convolution from 
(a6)^^~^aa;"+^~^^6a;""^^~^^ has a factor {ahf^. Hence every trans- 
vectant in (VIII) either vanishes or is the transvectant of a form 
having a factor (ab)^, with another form : and hence P can be 
expressed as a sum of terms each of which has a factor {aby\ 

A convenient way of expressing this is to write 

P = 0, mod {aby\ 

61. Owing to the large number of symbolical products which 
represent covariants.of given degree and order it is important to 
obtain methods of classifying them. For this purpose the greatest 
index of any determinant factor in the symbolical product is 
chosen. We shall use the word grade to denote this index. If 
the symbolical product is a covariant of a single binary form, 
then, by the theorem just proved, co variants of odd grade may be 
expressed in terms of covariants of higher even grade. On this 
account the German equivalent Stufe is used for half the index. 
We prefer to define grade as the index itself, since the classifica- 
tion is useful when the symbolical product is not merely a 
covariant of a single form. 

62. Covariants of degree 3. We proceed to obtain criteria, 
by means of which it may be at once determined, whether or not 
a given covariant of degree 3 can be expressed in terms of 
covariants of higher grade. These were first obtained by Jordan* 
in 1876. They were independently discovered by Strohf; and 
his method of investigation is given here. 

We consider first the covariants of weight w which are linear 
in the coefficients of each of three binary quantics 

/i = a,r^ , /2 = K""' , fs = c^"'- 
Let us write Ui for (be) a^, Wj for (ca) b^, m., for (ab) Cx ; then 

^1+^2 + ^3= 0. 

* Liouville, 2 S6r. in. 1876. 

t Math. Ann. Bd. xxxi. p. 444 et seq. 



60-62] gordan's series 63 

Hence 

v^a-iui = (- Vf-i m/ (w2 + Ms)^"^ 

^=" (?) 
Multiply this expression for u-P~^u^ by 

where h-^, k^ are any positive integers, and take the sum of each 
side of the result from i = Q to i = k2: hence 

= (- 1)^^+^ i^ (^) (^ ~ ^ ~^' ~ ^) t^/--t.3^ t (XIV). 

Now when 

g — ky> X > g — ki — k^ — \y 

therefore the last sum written down may be divided into two 
series, viz. \ = up to g-k^-k^-l, and \ = 5r - A^i up to ^. 

Let g — ki — k.2—l=k3. 

Then the first of these series is (writing i for X) 

In the second series, write g —i for X, then it becomes 

lf^)r'-t'~^U"-v (XV). 

* ( ) is the coefficient of x^ in the expansion of (1 + a;)'. 

t For S (^T^) ( " \~^) is the coefficient of x*» in (l + ar)^"'*' . (l + x)-^'-l. 



64 THE ALGEBRA OF INVARIANTS [CH. IV 

Now if I and m be positive integers 

\m) ^ ' m\(l-iy. ^ ^ \ m ) 
'l-\-m-V 



= (-Y , 
^ ^ V ^-1 

-(-r^'-'{~P_~i^) (XVI). 

Hence the series (XV) becomes 

|„<-)^"*-'(f)(t-~i>''""'^' 

=<->'•"■!„(?) (t- /)'"'"'<"■ +'^>' 

But the coefficient oiv^^~^Ui^ is here equal to 
^ (9\ (9 - ^\ (- h - 1' 



\,\J \i — X./ V ki — i 
^ (9\ (^1 + A^s - \ 



Hence the series (XV) becomes 

By means of the relations (XVI), the series on the left-hand 
side of (XIV) becomes 

Hence the equation (XIV) may now be written 

+(->'• I.© f'^fe "')"=""'"''=" <^™>- 



62-63] gordan's series 65 

This equation is true for all positive integral values of k^, k.,, k.^ 

for which 

k^-\-ko,-\-hi = g - 1. 

63. Let us suppose that w, the weight of the covariants under 
consideration, is not greater than the order of any of the three 
quantics. 

Then any one of the covariants may be expressed in terms of 
the set 

where t = 0, 1 , ... w. For since n.^ -^ w, whenever a factor (ca) 
appears in a covariant of this weight, it may be removed by 
means of the identity 

(ca) hx = — {he) ax — {ab) c^. 

Similarly all the covariants may be expressed linearly in 
tei-ms of the set 

(6c)"'-' {cay a/''-* hx''^-^'-'^ c/''""', 
or of the set 

(ca)"' - '■ {ahy a^'^ - *" h^'^- ' » c/3+« - w_ 

We shall suppose the members of each of these sets arranged 
in order according to increasing values of i. Then the forms of 
any one set are linearly independent. For suppose a relation to 
exist between the forms of the first of the above sets. Let the 
terms in this relation be arranged in the order indicated. The 
relation still remains true if we take for fy,f 2, fa special quantics 
instead of general ones. We shall suppose them to be merely 
powers of linear forms ; the result of this is that the letters 
a, b, c are no longer purely symbolical. Hence if 

Xi {ab)""-' {bcY aa,«'+^-«' i^;"^-"' c^,-"'- ' 

be the first term, we may divide the identity by {bey. Every 
term of the quotient except the first contains {be) as a factor; 
but the quotient must be zero, hence the first term must vanish, 
when we make b = c, 

ie. (aby-'-' aa;'*'+'--«' ft^w^.+ws-w'-' = 

which is clearly untrue. Hence no linear relation can exist between 
the forms of one set. And therefore there are exactly w +1 linearly 
independent covariants, which are of the first degree in the co- 
efficients of each of three quantics, and of weight w (:f> 7ii, Tig or n^). 
G. & Y. 5 



66 THE ALGEBRA OF INVARIANTS [CH. IV 

64. Writing in (XVII), for Ui, u^, u^ their values in terms of 
a, b, c, we obtain a linear relation between the first ki + 1 members 
of the first set, the first A^a + 1 members of the second set, and the 
first ks+l members of the third set, where 

ki + k2 + k3 = g — I =tu — 1. 

Thus we obtain a relation between w+2 forms. 

Hence if we take the first irii forms of the first set, the first niz 
of the second, and the first n^ of the third, where 

nil + rrii + nis = w + 1, 

we have a set in terms of which all other of these covariants 

can be expressed. This set may be chosen so that it contains 

2w 
no covariant of grade less than -^ . 

For if w = 3m — 1, we may take 

and we see that all covariants can be expressed in terms of those whose grade 
is 2m at least. 

If w=Zm - 2, we may take 

and we may express all covariants in terms of those whose grade is 2?h - 1 at 
least. 

If w=3m. we take 

mi = m + l, m.2 = 7n^ = m 

and we have to include one covariant of grade 2m, the rest being of grade 
2m +1 at least. 

In the second case there is one relation between the 3m covariants whose 
grade is <|;2m— 1 ; viz. by (XVII), 

(a6)2»» - 1 (6c)"* - 1 aj;»i - 2»* + 1 b^'h - 3m + 2 ^^..3 - m + 1 

4- (6c)2"» - 1 (ca)"* - 1 a A " »» + ' 6a;"* ~ "'"" * '■ t'x"' - 3"' + 2 

+ (oa)2»» - 1 (a6)'»~J aj:"i -3m + 2 ft^Tiu - m + 1 g^H, - 2», + 1 

= 2C^ , (XVIII), 

where C'2,,, denotes a covariant whose grade is <t:2»!. 

In the third case there are two relations between the covariants of grade 
■^2w ; these will be found from (XVII) to express that the difference between 
any two covariants of grade 2tn and weight Sm, 



64-66] gordan's series 67 

It would be sufficient for the general theory to prove the 
theorem : — If Uj + it, + Us = then there are w + 1 linearly in- 
dependent products of u^, U2, u-s of order w such that each contains 

an exponent of — at least : this is not easy. 

65. Next let lu be greater than the order of one or more of 
the quantics. 

If iv>ni, the sum of the indices of (ab) and (ac) cannot be 
greater than n^, and hence (6c) must have an index equal to 
w — III at least. 

We define the quantity e,: to be w — Ui if w > ni, and to be 
zero in the contrary case. Then each of our covariants of 
weight w must have a symbolical factor 

{bcy^ {cay^ {aby\ 

The remaining factor will be a symbolical product representing 
a CO variant of weight 

W — €1 — e.) — €3 = •57, 

of the quantics 

^^M,-e.,-e3^ J^Mj-ea-fi^ p^^Ms-n-fj. 

But the weight or is not greater than the order of any of 
the three quantics : hence we may apply the results of the last 
paragraph. That is, all the covariants of weight w may be 
expressed in terms of ct + 1 of them. If ei, e.^, €3 are unequal 
our choice of covariants in terms of which the rest are to be 
expressed may not be the same as before. But the student will 
have no difficulty in writing down the result as regards grade. 

Ex. Prove that any covariant of degree three can be expressed in terms 

of covariants of grade — .^ — at least. 

66. We have now two different series which may be used for 
obtaining relations between covariants involving three symbols. 
These are series (IX) due to Gordan, and Stroh's series (XVII) 

( - 1)*-^ % rj) (^' '^^' ~ *) (ab)^ - '■ (bey a^''' - «>+' 6/^ - '" c/^ ->...= 0, 

where ki + A'a -f- A^s = w — 1 

and w is not greater than any one of the numbers 

r'l, ?'2, n.,. 
If this condition as regards lo be not satisfied we write 
w — til = 61, w — n., — e.i, w — ??3 = fj 

5—2 



68 THE ALGEBRA OF INVARIANTS [CH. IV 

where it is understood that 6 = if n>w. Then the reduced 
weight ta- is 

W — €i — 62 — 63. 

In this case Stroh's series is obtained from (XVII) by writing 
■53- for w, Wi — €« — 63 for )h, n^ — ea-ei for n.^, ns-Ci — e^ f'^r "s, 
and multiplying the result by 

(6c)'i (ca.y^ (abyK 

Here k^, k^, k-s satisfy the relation 

^'1 + A^a + k^ = 'S7 — 1. 

The advantage of Stroh's series is that it gives all possible 
relations between the covariants under discussion. It is, however, 
generally more convenient to have relations between transvectants 
than to have them between symbolical products. Thus although 
series (IX) does not give all possible relations, yet it is frequently 
the more convenient one to use. 

By means of series (VII) and (VIII) we may translate Stroh's 
series into a relation between transvectants. In fact this is what 
Stroh himself does. This relation has the disadvantage that the 
coefficients in it are themselves series. 

It is convenient to have a short method of referring to Stroh's 
series ; we therefore introduce the scheme used by Stroh 

/i 72 fa 

tti rCft fC^ 

which is distinguishable from Gordan's scheme by the weight 
being indicated outside the bracket. 

67. The quantics of low order furnish very few examples of 
covariants containing three symbols concerning which Gordan's 
series gives incomplete information. We have mentioned (f, H)^ 
as one such case. The co variant ((/, if, /)'' of the sextic / — 
where i = (/,/)* — is another. The reader will have no difficulty 
in proving that 

Ex. Prove that the covariant 
of the binary form ax^^ = bx^^ = Cx^'' 



66-69] gordan's series 69 

can be expressed in terms of covariants of higher grade ; and that the 

covariant 

{ahf {acf (bcf W^ c^^ 
vanishes identically. 

68. Let the three quantics /i,f-2,f3 be made identical, so that 

Then if lu ::j> n, all covariants of weight w and degree n can be 
expressed linearly in terms of those whose grade is not less than 

2w 
3 ■ 

If lu = 3w — 2, it is possible to go a step further ; we may 
express all these covariants in terms of such as are of grade 2m 
at least ; since covariants of a single quantic of odd grade may 
be expressed in terms of covariants of higher even grade. 

Hence if w = Sm— 8:if'-n, 8<3, all covariants of degree three 
and weight w can be expressed linearly in terms of those whose 
grade is 2m at least. 

Similarly if w> n, we have 

ei = €-2 = €s = € = w — n, 

■S7 = w — 'Se. 

Then we may express all covariants of degree 3 and weight w 

in terms of those whose grade is not less than —- + €=-- — e. 

o o 

If € is odd the lowest grade given by this is odd unless ct = Sm — 2, 

in which case we see on multiplying (XVIII) by 

(aby (bey (cay 

that we may express covariants of grade 2m— 1, in terms of 

covariants of higher grade. Hence if e is odd, all these covariants 

2w 
may be expressed in terms of those whose grade -^ — — e + 1. 

If e is even, then as before we take tv = 3m — S, 8<S, and the 
minimum grade becomes 2m — e. 

69. There is one further point in this matter to be noticed. 
It has been proved that all covariants which are linear in the 
coefficients of each of three given quantics and are of given 



70 THE ALGEBRA OF INVARIANTS [CH. IV 

weight w can be expressed in terms of those whose grade is 
higher than a certain number. Further it has been shewn that 
among the covariants actually retained, no linear relations can 
exist. We passed to covaiiants of a single quantic, and deduced 
that all covariants of degree 3 and weight iv can be expressed in 
terms of those whose grade is higher than a given number. Is it 
possible that amongst the covariants retained here, there may 
exist other relations which do not appear in the general case ? 
To answer this question we express the symbolical products as 
transvectants by means of § 53. The product 

(ab)'^ - *■ (be)' ax"'+' " '" bx''^ ~ '" Ca;"» " ' 

where as before C,„ indicates a covariant whose grade is not 
less than m. 

The covariants retained will be — when w i^ n — 

{(fuAYjsr, {(AfsYj.f, ((/3,/)»,/.y 

where a ^j; -^ . If ?(; = 37/i — 2, there will be one relation : and if 

o 

w = Sm there will be two relations between these covariants. 

Let us suppose all the quantics to become identical. Then 

those covariants for which a is odd vanish. Let us suppose 

2'W 
that amongst the remaining covariants for which a-^ -q- a relation 

exists, say 

Then using Aronhold's operators which may be written for 
short (/ ^j , (/2 ^j , (f, ^j we obtain 

— since Oi is supposed even. 

Hence corresponding to a relation between the covariants for 
a single quantic, we may deduce a relation between the corre- 
sponding covariants of three different quantics. The results 
obtained then, for a single quantic, are as complete as those from 
which they were deduced ; and no linear relation can exist between 
the retained covaiiants. 



69-72] gordan's series 71 

70. The covariant 

(«6)^ (bcY (cay aa;''-"-^ b^'''^-'' Cx"~''~'' 

of the binary form a^^ = b^^ = Cx\ can be expressed in terms of 

n 
covariants whose grade is greater than \, provided that A- ^ ^ 

and IM + v> -^; unless 

\ = ^ = 1/ = - . 

The verification of the above important theorem is left to 
the reader; it is really only a restatement of the theorem of 
§68. 

It should be noticed also that a similar theorem is true when 
the letters a, b, c do not refer to the same quantic ; it is 

{aby- (bey (cay aa;'*'-"-^ bx'"'"^''' c^^-*"-" = S C^+i, 
provided that fx + v>^. 

71. Ex. Prove that the following covariants of f=aj^ vanish 
identically : 

((/, ff\ ff " ^ when n = 4A, 4X - 1, or 4X - 2, 

{{fjf\fT when n = 4X + l, 

{{fjf^^'\fT~'^ when 7i = 4A + 2. 

And shew that no other covariants of degree three vanish except those 
included in the form 

U.ff''^\ff- Stroh. 

72. Covariants of degree four. It is the object of the 
present paragraph to determine the conditions that a covariant of 

degree four and grade \ ( where X,:|> - J may be expressible in the 
form 

n n n 

2C;,+, + (a6)2(6c)'2(ca)2.^: 

the expression C^+i denotes — as before — a covariant of grade not 
less than X, + 1, and ^ being a covariant of degree one can only 
be the quantic itself. Of course the second term cannot appear 
if n is odd. 



72 THE ALGEBRA OF INVARIANTS [CH. IV 

It has already been proved that any covariant 

C = {ahy (bcY (cay a^''-"-^ b^''-^-'^ Ca;"~''~^ 

where fM + v>-^, is of grade gi-eater than \, unless 

Auy covariant obtained by convolution from G is of the same 
form as C, but the indices of some of the determinant factors are 
increased ; hence any covariant obtained by convolution from G 
either is of grade greater than X., or else is the invariant 

n n n 

(abf {bcf (caf . 
Any transvectant (G, Fy, where F is any binary form, is also of 
grade greater than X or else, if p = and \ = fj, = v = -^, has a factor 

n n n 

(a6)2(6c)2(ca)2. 

Further, any term of this transvectant differs from the whole 
transvectant, by a linear function of transvectants 

{c,Fy', 

where C and F are obtained by convolution from G and F. Hence 
any term of a transvectant {G, Fy is equal to 

lG,+, + (abf^(bcy(ca)K^, 

where as before G^^+i is used to denote a covariant of grade X + 1 
at least. 

Again, any covariant having a symbolical factor 

(abY (bcY (cay, 

where /x, + y > ^ , is a term of a transvectant (G, Fy, and hence may 

be expressed in the form 

SC;,+i + (a6)2(6c)2(ca)2.^. 

Consider now the covariant of degree four where v If-X 

K = (abf (bey (cdy aa,""^ bx''-^-" c^"-''-'' d^'"". 






gordan's series 73 



If /i > - , then 

K= tC^+, + {ahf {hcf {caf . ^. 
Otherwise by means of the relation 

{cd) ax = (ad) Cx — (ac) dx 
we obtain, since ^ <t: X <^ i^, 

^ = 2(-)<(;)£„ . 
where 

Li = (abf (bey (ac)'^^ {adf aa,"-^-" hx''-^-'^ c^'-i'-'^^ dx"^. 

If either 

. \ 

or 

. \ 

then 

Li = SC,+, + (ab)hbc)hca)i ^. 
But if 

yu. + 1/ > X, 

one of the above inequalities must be satisfied ; hence in this case 

n n n 

K = 20x+i + {abf (bey (caf . ^. 

Next consider the most general symbolical product ^ of degree 
four ; it is sufficient to write down its determinant factors, which 
we take to be 

(aby (acY' (bcY' (ad)"' (bd)"^ (cd)"*. 

It is supposed here that no index is greater than X, which is 

n 
itself not greater than ^ . 

By means of the identities 

(cd) ax = (ad) Cx — (oa)) dx, 

(cd) bx = (bd) Cx — (be) dx, 

we can express K in terms of covariants L in which either (i) the 
index of (cd) is zero, or (ii) the indices of both ax and bx are zero. 



74 THE ALGEBRA OF INVARIANTS [CH. IV 

The indices for the covariants L will be denoted by accented 
letters. In the second case 

X-\- fj^ -\-Vi =11, \ + fi^' + V2' = n, 
hence 

fh' + fjh+ vi + v^ = 2?? - 2\ ^ 2X, 

consequently either 

/a/ + /a./ > 2 or i/j' + j/g' > 2 • 
Therefore 

K=XL= SOx+i + {ahY {hcf {cay ^. 
In the first case 

A^l' + tl2 + Vi + I// = /tti + /*2 + ^1 + ^2 + J'S, 

and if this sum is greater than \, the same result is true. 
Hence — Wken fiy + n^-v Vi-\- v^-\- Vz> \ 

the covariant of degree four ivhose determinant factors are 

(abY (acY^ (bcY' (adY' (bdY' (cdY\ 
may be expressed in the form 

XG,+, + {aby(bcf{cay^. 

73. Any covariant which contains the symbolical factor 

(abYibcYXcdY, 

n 
where /a + j/ > \, and X:lf> ^, may be expressed in the form 

tG^+. + iabf'ibcf^cay'^. 

For if r be such a covariant, and 

K = {abY (bcY (cdY «x''~^ &/"^"'' c,^"-''-'' dj'-" ; 
then _ _ 

Each term of this sum has just been proved to be of the 
required form. 



72-76] gordan's series 75 

74. Any covariant of the covariant 

Ar^2n-2>- = ^ahy a^"-'- 6^"-'-, (7- :|> |^ 

of the binary qumitic f= Ux^ = bx' = ..., may be expressed in the 
form 

n n n 

tC,+, + (ab)HbcY(cay^, 

where Cr+\ represents a covariant of grade not less than r-\-\. 

To prove this we observe that any form obtained by convolution 
from a product 

{ahy ax''-'' bx''-'' {cdf Cx''-' dx""-"" ■ . . 

is either of grade greater than r or else has a factor of the form 

(aby{bcy(cdy. 

In the latter case by § 73 this covariant may be expressed in 
the form 

71 n V 

^Cr+^ + {aby'(hcy{cay.^. 

Now any covariant of kx^"~'-^'' may he expressed in terms of 
transvectants of the form 

({k, ky, n. 

But the transvectant (k, ky is a linear function of covariants 
of / obtained by convolution from 

{aby ax''-' bx""-'' . {cdy Cx''-'' dJ'-\ 
Hence the theorem is true for covariants of the second degree ; 
and therefore for all covariants of kx^'"^''. 

75. It is well to notice that nowhere in the last three 
paragraphs has it been assumed that two different symbolical 
letters refer to the same quantic. The theorems are thus true 
when some of the letters refer to different quantics. For example 
the theorem of § 74 is true for covariants of 

lgm-'2r ^ (^jy ^^n-r J^^n-r 

when the quantics ax', bx" are different. 

76. The discussion of covariants of degree four ma}' be carried on a step 
further. 



Thus if X is even, and 



fi + p=\:^-, 



76 THE ALGEBRA OF INVARIANTS [CH. IV 

then the covariant 

K= {ahf {bcf (cd)" a^"- Hj''^-" cj' " '^ " " tf^" " " 
differs from 



2, 



by an expression of the form 






To prove this we notice that the index of 6^ in ^ is » - X - /x, and this is 
not less than v ; for by the inequalities above we see that 

hence we may use the identity 

(cd) hx = {bd) Cx - {be) dx 



to obtain 



A--2(-1).Qa 



where 

Now if 

. X 

• ^ 

or ii + v-i>-, 

then A=SC\+i. 

(The weight of ^ is 2X ^« ; hence no term with the factor 

n n n 
{ab)^{bc)^{cayi 
can appear.) 

Then the only term we need consider is 

X A _^ _6 

L, = {abf{bcfibdfaJ'-'bJ'-'^C''d^'''\ 

2 

Again, let K' be that particular covariant of the form K for which 
^=0,v=\. Then 






76-77] gordan's series 77 

Hence 

In just the same way it may be proved that if X is odd, and 



(1) f. + v=\:^-, 



then K=2C\^^ : 



n 



or (ii) /x + v + l=X j>-, 

then 



X- 1 
X-1 



2 

The results of this paragraph may be stated as follows : 
(i) IfXbe even and fi + v = X, then the covariant 
{aby {bcY (cdy a.,."-^ 6^;"-^-'" c,."-'^-'' c?.,."-" 

can be expressed as a sum of a reducible covariant and covanants 

of grade greater than \. 

(ii) If \ be odd and fi+ i' = X, then the above covariant can be 
expressed as a sum of covariants of grade greater than \. 

IfXbe odd and /ji + v = X—'l, the covariant can be expressed as 
a sum of a reducible covariant and covariants of grade greater 
than X. 

Ex. (i). Any covariant which contains the factor 

{abf {acp {bc^' {ady (bd)"' {cdY\ 

where ^i+/x2 + j'i + i'v, + i'3>X and X<-, 

can be expressed linearly in terms of covariants whose grade is greater 
than X. 

Ex. (ii). Prove that if a covariant C of degree 5 has the factor written 
down in the last question, for which 

/ii + /H2 + «'i + i'2 + i'3=X, and X:f»-, 
then C='2,C^,^+ reducible terms. 

77. Jacobians. The first transvectant of two binary forms 
fx^> 4^x^ is called their Jacobian. It is, in fact, equal to 

mn d {Xi, Xj.) 



78 



THE ALGEBRA OF INVARIANTS 



[CH. IV 



The following properties of Jacobians are important. 

(i) If f, (f>, y^ he three binary forms, each of order greater 
than unity, the Jacobian of the Jacobian of f and <p with y^ is 
7'educible. 

Let /=tta;^ <f> = b^\ yir^c^P 

{f<\>) = {ab)a^^-'h^''-\ 

Polarize once with respect to y, the result is 

^■^' ^^^ = ^/i^^ ^"^^ ''"""' ^^"~' *^ -^ m + n--"2 ^"^^^ '*^"" ^^"" ^^- 

Hence 

(m + n-2)((/,<^),^/r) 

= {m - 1) (a6) {ac) a^""-- b:^"'' c/-^ + (?i - 1) (ab) (be) a^"'-' 6/-- c/"'. 
But 2 (a6) (ac) ^j^; c^ = (abf c^^ + {acf 6^' - (60)^ a^^^ 

and 2 (a6) (6c) a^ c^, = - {ahf c^ - {bcf a^' + {acf b^? *. 

Therefore 



= iaa;'""'6a,'*-''c/- = 



m — n 
w + n — 2 



{aby c/ + (acY b^' - (be)- a^' 



m — n 



-^^^^^_^{f<f>r.ir+hAffr.<f>-H<t>,i^y'f-{^i^)- 

(ii) The product of two Jacobians may be expressed as a sum 
of products of covariants, ther^e being at least three covariants in 
each product; provided the forms of luhich the Jacobians are 
taken are all of order greater than unity. 

To prove this, we first establish a useful identity between 
symbolical forms. 

Consider the determinant 



(/,- 


Wl«2 


ai 


b{^ 


bA 


b,' 


c,' 


C1C2 


Cf 



til «9 



it vanishes if -p = r . hence (ab) is a factor ; similarly [be), (ca) are 

factors. There can only be besides a numerical factor, which may 

* See Chapter I. § 22. 



77] 



GORDAN S SERIES 



79 



be determined by considering the coefficient of a^hih^c^. The 
determinant is therefore equal to 

— {ah) (be) (ca). 

Hence 2 . (ab) (be) (ca) . (de) (ef) (fd) 

di - 2d,d. di" 
62' Z6162 61 

fi -2/,/, y,^ 

(adj (aef (aff 

(bdy (bef (b/y (XX). 

(cdf (cey (c/y \ 

In this identity let us put 

Ci == sCo, C2 = ^1 5 y 1 ^ '^■2> j2^ ^i- 



a^' 


a^a^ 


ai 




w 


bA 


bi 




c,' 


C1C2 


ci 





Then 



2 (ab) ax b^ . (de) dx e^ = 



(ady (aey Ox^ 

(bdy (bey bx' 

dx' e^ 



Consider now two Jacobians (f, (j)), (yjr, -)() ; where 
/=a^- = 6^", 



Then 



(/ 4>) ■ (^. %) = (a^) ax'""-' ^x'-' . (de) dx^-' e^'"^ 



= \ax'^-'bx''-'dx^-'ex'i-' 



(ady (aey a^ 

(bdy (bey bx' 

dx- ex- 



+ i(<^.t)yX-i(0.%)V-f (XXI). 

For the sake of generality the forms /, <^, i/r, ^ have been 
supposed to be all different. The theorem is still true if two are 
equal to one another. In particular, it is true for the square of a 
Jacobian. 



80 



THE ALGEBRA OF INVARIANTS 



[CH. IV 



Another symbolical identity of interest is obtained thu-s : Form by means 
of the ordinary law of multiplication the product of the following vanishing 
determinants : 






ajttg 


ai 







Wh 


V 







CiC.2 


«•/ 







d^d.^ 


di 








Se^ea 







fi -2/,/2 f^ 
9% -'^9x9% 9i 



{agf (ahy 
(Jbgf {bhf 
{cgf {chf 



= (XXII). 



Hence if 



= 0. 



{aef {a/y 

{bey {hfy 

{cef {c/y 

{def {dff {dgY (dh) 

F=e^'', *=//', ■9=g/, X=A/ 

i (/, Ff (/, *)2 (/, •^f (/, x)2 

^ {<i>,Ff ((^,*)2 (0,<?)2 (0,x)2 

(v^.i^)-'' (>/.,*)i' (>/.,*)2 {^,xy 

U,Ff (;^, *)2 (;f,*)2 (;t, X)^ 

Here, as before, the forms are not necessarily all diflFerent. 

78. The expression for the product of two Jacobians may 
also be obtained as follows. 

We have 

(/ «^) X i-^, X) = («^) dx'^-'ho,''-' • {de) d,P-'e,9-' 

= (ab) (ae) a/^-^ft^^-^e^?-! . ^jr - {ah) {ad) a^'^-^b^^'-^d^P-' . x- 
and, by means of the identities of the type 

{ab) {ae) b^e^ = {ab^ e^ + (ae)'- bi - {hef ai, 
the right-hand side becomes 

\ (/ xr <f>f + ^ i<f>> n/x - i (/ ^)^ <t>x-i (4>, xT-ff 

as before. 

Thus if J be the Jacobian of y and 

- 2J^ = {fjy <f>^- + (</,, cf>yf^ - 2 (/, <f>)\f<i>. 

79. Copied forms. If in the symbolical expression of a 
covariant 11 of a binary form F^^, the symbolical letters are taken 
to refer to another binary form <f>x^ where m^p, and 11 is multi- 
plied by 

aJ''-PbJ"-v ... 



77-79] gordan's series 81 

— where a,b, ... are the letters occurring in H — then the resulting 
form IT' is called a copied form. The original form 11 is called 
the model form. 

Ex. The Hessican of any binary form (l> = a^^^ = hx^^ is 
(a6)2a;,'»-26;,'»-2. 
It is formed on the model of the Hessian of the quadratic. 

The use which we are going to make of the idea of copied 
forms is for the case when the orders p and m of the fundamental 
quantics of the two systems are the same. The form ^ will be 
taken to be a covariant of degree two and order m of a binary form 
/x'*: then 

<^ = (/, ff<^ = {ahf" a^""--" ba,''--" = (f>x^-*^, 2n-4ia = m. 

On the model of the complete system for the general binary 
form F^^, we may construct a complete system for (f). If when 
this is done the symbolical letters <^i, <f)2, ... which refer to (f) are 
replaced by the symbolical letters a, h, ... which refer to yj each 
covariant of </> will become a covariant of/, which consists of a 
sum of sj'mbolical products instead of a single term. These 
separate products may be arranged in a series of adjacent terms, 
the difference between any two of which contains a factor of the 
foim (a6)^'^"'"'. Thus if we reject covariants of grade greater than 
1(T, any one of the terms given by a covariant of ^ may be taken 
to represent this covariant. Before proceeding to a rigid proof of 
this statement, let us consider an example. 

The covariant 
of the sextic 

"z — "x — • • • 

is a quartic. Let us consider the Hessian of <^, 

It consists of a linear function of ten covariants of the following tj'pes : 

{abY{cdf{acfb^^d^\ 

{ahy{cdY{ac){ad)h:?c^d^, 

{aby (cd)* (ac) (bd) a^bxCxd^. 

Any one of these may be taken for the copied form, for each differs from 
the whole transvectant by transvectants of forms obtained from 

iab)*ax^bx% [cdfcx^d^^ 
by convolution. Hence when forms of grade higher than 4 are neglected, the 
whole transvectant and each of its terms are equivalent. 

G. & Y. 6 



82 THE ALGEBRA OF INVARIANTS [CH. IV 

80. Consider any symbolical product P, the symbols of which 
refer to 

in this product let us replace any particular letter <^<^' by a new 
variable y ; (we replace ^12 by y^ and ^u by — ^o). Let Py^n-w be 
the resulting expression, then 

= (Pj,2n-4<r , {aVf^ ay''-'"' by"-'"'y"-^ . 

Any term of this transvectant dififers from the whole trans- 
vectant by terms involving a factor (aft)"^"'"'"' : for since y is absent 
when the transvectant is expanded, the only kind of adjacent 
terms are of the form 

(aa) (y86) M, (ab) (/3a) M (see § 50). 

The above argument is not affected if we suppose that 
symbolical letters are present in P which do not refer to <f>. 
Hence we may replace each of the letters which refer to (f> in turn 
by letters which refer to /; and in doing this we may at each 
stage choose any one term to represent the whole expression P, 
provided that those terms which involve (aft)-""*"' are rejected. 
Thus taking the quartic invariant {abf(bcy(cay for model, we 
obtain the copied invariant of 

<\) = {abya^^b^\ 

viz. {(f>M^ i^ifk^iy ((t>3<f>iY- 

Any one of the sextic invariants 

(aby(cdy{efy(acy(dfr{eby, 

{aby(cdy{efy{ady{cey(fby, 

(aby (cdy (e/y (ac) (ad) (ce) (de) {/by 

differs from the invariant of (f>, by terms involving the factor {abf. 

Conversely in any co variant of a binary form aa;"= 6/'= ..., 
which has a symbolical factor (ab)-'', we may replace the letters 
a, b wherever they occur in the symbolical product by a single 
letter <f>, and remove the factor (aby'^ altogether: on the under- 
standing that forms of grade greater than 2a are being rejected, 
and that refers to the form (a6)-'^aa;"~-'^6a;"~-^ The reader will 
have no difficulty in verifying this statement. 



80-82] gordan's series 83 

81. G-eneralized Transvectants. Consider a product 

of any binary forms, 

/i(^) = ax'", My) = hy'\ fM = o^P. 

The result of operating on P with 

{m-\-^l) \ {n-\-v)\ (p-^^iiOUx Of^ O^ 
m\ ■ n\ 'pi ^^^y^^^z^^z,, 

where 

n.,= ^ - ^ 

*■' dx-idy^ dx^di/i ' 

after operation y and z being replaced by a;, is 

(aby {acY (chy aa;"*-^-'*6a;"-■^-''Ca;^-'*"^ 

This we define to be a generalized transvectant. 

Instead of taking forms each having a single symbolical letter, 
we may construct generalized transvectants of forms each of which 
has two or more symbolical letters. Thus we may replace a^;'" by 

The generalized transvectant may then be expanded as a linear 
function of certain symbolical products. Just as in the case of 
ordinary transvectants, any term of a generalized transvectant 
differs from the whole transvectant by lower transvectants of 
forms obtained from the original forms by convolution. We leave 
the verification of this statement to the reader. 

Further it is evident that any symbolical product may be 
regarded as a generalized transvectant. A copied form is then 
merely the same generalized transvectant with a new form taken 
for ground form. 

The theorem of § 80 — that any single term of a copied form, 
when a covariant of the second degree is the new ground form, 
may be taken to represent the whole form, provided that forms 
of higher grade than that of the fundamental ground form 
are to be neglected — is merely a particular case of that just 
enunciated. 

82. Hyperdeterminants. When the forms of a generalized 
ti-ansvectant are all the same, it will be noticed that the trans- 

6—2 



84 THE ALGEBRA OF INVARIANTS [CH. IV 

vectant is entirely given by the differential operator. Thus a 
covariant of the binary form /= ax^, is completely defined by the 
operator 



^^ xtj i^ xz ^''"zy 



Cayley used the notation 

to define such a covariant. These symbolical forms are called 
hyperdeterminants. Cayley introduced his calculus of hyper- 
determinants some years before the symbolical notation was 
invented by Aronhold. 

The hyperdeterminant notation was introduced for a single 
binary form merely for convenience. It is evident that it may be 
used perfectly well for covariants of two or more different forms. 

It is interesting to notice that the letters of a symbolical 
product may be regarded as differential operators. Thus if 

_ a p _ a_ _d 

"'-d^,' ^^"a.7i' '^^~afi' 

_ a „ 8 d 

dfa 07]^ 0?2 

then {a^Y (ajY (7/3)" 

operating on the product of 

produces the covariant 

{ahY {acY (chY tta:"*"^"'* &«""*"" Cx^*""" 
multiplied by 

ml 111 p\ 

(m — \ — fi) \ {n — \ — v) I {p — fj, — v)\' 

provided that after operation ^, rj, ^ are each replaced by x. 
Further the operator 

(a^Y (<^yY (j^y olT-^-'^ /S^"-^-" 7xP-'^-'' 
acting on the same product produces the same covariant multi- 
plied by 

m\n\p\. 

In this case ^, 17, ^ all disappear after operation, so there is no 
question of replacing them by z. 




CHAPTER V. 

ELEMENTARY COMPLETE SYSTEMS. 

83. Complete Systems of irreducible covariants. We 

shall devote this chapter to a detailed discussion of the invariants 
and covariants of single binary forms of the first four orders ; in 
particular, we shall obtain what are known as the complete 
systems of covariants for such forms. It has been observed, in 
fact, that the symbolical notation enables us to construct an 
infinite number of covariants of any form y*, but, as was first 
proved by Gordan, all these are rational integral functions of a 
finite number of covariants of f\ this finite number is said to 
constitute the complete system of irreducible concomitants, or, 
more briefly, the complete system of concomitants of the form. 
The general proof of Gordan's Theorem will be given in the 
next chapter. For the present we shall content ourselves with 
explaining easier methods of obtaining the complete systems in 
the simpler cases, and proving of course that such systems are 
actually complete. 

Inasmuch as every covariant can be expressed as an aggregate 
of symbolical products, we need only retain such as consist of one 
product in seeking for the complete system. 

84. Linear Form. The discussion of a single linear form 

/=a* = &x = etc. 
presents no difficulty. 

For a symbolical product either contains a factor of the 
type (a6) or it does not. If it does so it is zero because 

(a6) = 

and if it does not it is simply a power of/. 



86 THE ALGEBRA OF INVARIANTS [CH. V 

Hence every covariant of a linear form is a power of the form 
itself, or in other words, the form constitutes the complete system. 

Go7\ The same argument applies to any number of linear 
forms, for every symbolical product is a rational integral function 
of invariants of the type (ab) and co variants of the type ag.. Hence 
the complete system for n linear forms consists of the n forms 
themselves and the ^n{n — l) non-vanishing invariants of the 
type (ab). 

This result has already been established in § 37 where it forms 
the lemma preliminary to the proof of the fundamental theorem. 

85. Quadratic Form. Suppose the form is 

/= cix^ = bx^ = Cx^ = etc. 

Then if a symbolical product contain no factor of the type {ab) 
it is a power of / ; if on the contrary it contains such a factor {ah) 
it can be transformed so as to contain {ab)- (§ 60), which is an 
invariant. Thus every invariant and covariant except 

^x, {ahy 

can be expressed in terms of co variants of lower degree, hence by 
continued reduction we infer that every such form is a rational 
integral function of 

/(/./)- A, 

the latter being the discriminant of the quadratic. In other 
words, the form and its discriminant constitute the complete 
system. 

Ex. Prove that 

{ah) (ac) {bd) (ce) dxOx = jAy. 

(By interchanging a and h put the factor {abY in evidence.) 

86. Before proceeding to the discussion of the cubic and the 
quartic we shall explain some general principles relating to the 
formation of the irreducible co variants of any given degree of a 
binary form. 

Suppose that the form in question is 

/= tta," = 6^" = c^'' = etc. 

then the only irreducible covariant of degree one is/ 



■ 84-cSfil 



84-86] ELEMENTARY COMPLETE SYSTEMS 87 

Next, the only covariants of degree two are those of the type 

(a6)'-a^'»-'-6^''^-''; r = 0, 1, 2, ... 7i. 

If r be odd this covariant vanishes, and if r be zero it is 
reducible since it is equal to f^; the remaining forms corre- 
sponding to even values of r constitute the complete set of 
irreducible covariants of the second degree. 

Now assuming a knowledge of all the irreducible covariants of 
degree less than m we shall shew how to find the irreducible 
covariants of degree m. 

Suppose that the given irreducible forms are 

then any covariant of degree less than m is a rational integral 
function of / and the ^'s. Now a covariant of degree m is an 
aggregate of symbolical products containing m letters ; let C^ be 
one of the products and k one of the symbols involved, then Cm 
is a term in a transvectant 

where G-„i-\ is a product containing only ni — 1 letters, that is, 
it is a covariant of degree m — 1. 

Thus 6',, = (6',„_: , fy + S (C^, /)"', p'<p 

and C',„_i is derived fi-om Cni-\ by convolution. 

But C„,_j, C,rt_i being covariants of degree m— 1 are rational 
integral functions of the forms 

/ 01, 0i, ••• 0r, 

i.e. they are aggregates of terms of the types 

f^m-i=/''0,'^' ... 0A 

of degree m — \. 

Consequently (7,„ is a sum of transvectants of the form 

(f^^n-i,/)", 

where of course p :|>- n. Since this is true for every separate term 
in a covariant of degree m it is true for the whole ; or, in other 
Avords, every covariant of degree m is expressible in terms of 
transvectants 



88 THE ALGEBRA OF INVARIANTS [CH. V 

where Um-i is a product of the form 

and is of degree m — 1. 

Hence to find all the irreducible covariants of degree m we 
have to write down all transvectants of the form 

and reduce as many of them as possible. The remaining ones are 
the irreducible forms of degree m, for any covariant of degree m 
can be expressed in terms of them and covariants of lower degree. 

For example, in the case of a binary quintic the irreducible 
forms of degrees one and two are 

f,H= (aby a^ b^, i = {aby a^ b^. 

The only products of powers of these of degree two are f"^, H 
and i, so that all the irreducible covariants of degree three are 
included in 

(/^/)^ {Hjy, {ijy, 

where p ::|» 5 for the first two transvectants and p :^ 2 for the third, 
since i is a quadratic. 

87. Let us now return to the transvectants 

{'Urn-.fy 

which we have to reduce as far as possible. That many of them 
are reducible follows from the following principles. 

Suppose that ^m-i = yW 

where Fand W are likewise products of powers of/, <^i, i/),, ... <^,., 
but of smaller degree than Um-\, and further suppose the order of 
W is not less than p. 

Then if T^ be any term belonging to V and Tn_ any term of 
the transvectant {W,fy, which is a possible transvectant because 
the order of W is at least equal to p, T^T., will be a term in the 
transvectant 

(Um-^jy. 

Hence 



{U,n-^,fy=T,T,+ t{[I,r^„fy■,p'<p 

and Ti, T^ being both covariants of degree less than m are 
expressible in terms of f and the ^'s. 



86-88] ELEMENTARY COMPLETE SYSTEMS 89 

Now in discussing the reducibility of the transvectants 

let us consider them in the order of their indices, e.g. we examine 
all those of index one before we proceed to any of index two, 
and so on. 

Then since Um-\ is a covariant of degree m — 1 it is an 
aggregate of products of the type Um-\ , and since p < p it 
follows from the equation 

that the transvectant on the left is completely expressible in 
terms of co variants of less degree and transvectants previously 
considered, or to put the matter briefly, it is reducible for it 
certainly cannot give rise to a new irreducible form. 

Hence the irreducible covariants of degree m can only arise 
from such transvectants 

{Um-^,fy 

for which Um-i has not a factor of order greater than p. Thus in 
the case of the quintic no transvectant of the type 

(A/y 

is irreducible because p 1f> 5 and the term f^ contains a factor / 
whose order is 5. 

In the general case if Um-i possess a factor W whose order is 
not less than n, then the product Um-i can give rise to irreducible 
covariants for no value of p ; it may therefore be neglected entirely 
in the search for irreducible covariants. 

As an application of this remark we note that if the order of 
one of the </)'s, say (f>p, be at least equal to n, then we need not 
consider the product of this form with any others. 

Finally if 0g be an invariant we may leave it out of account 
in forming the transvectants, because it would occur as a factor 
in each transvectant in which it appeared and so the transvectant 
would be reducible. 

88. Irreducible system for the binary cubic. After 
these preliminary explanations the deduction of the complete 
system of a cubic presents little difficulty. 



90 THE ALGEBRA OF INVARIANTS [CH. V 

Let the form be 

f=a^^^ hi = d = etc. 

then the only irreducible form of degree two is 

H={ahya^K = hi = h'i. 

To find the irreducible forms of degree three we note that the 
product /- is negligeable, so the only possible irreducible forms are 

{H,f), {Hjy. 

Now 

(H,f) = {ahy{ac)hci = -t, 
where t is an irreducible covariant, and 

{H, fy = {{aby a^ h, c^]' = (aby (ac) (be) c^ 
= — (be) (ca) (ab) {(ab) Cx} 
= — ^ (be) (ca) (ab) {(ah) c^ + (be) a^ + (ab) Cx], 
as we see by interchanging a, c and b, c and adding the results. 
Hence (H,/)'- = and the only new irreducible form is t. 
The prodvicts of degree three are 

/'. a/, t, 

of which the first two may be neglected ; hence the irreducible 
forms of degree four are included in 

(tj), (t,f)\ (tjy. 

Of these (t,f) is the Jacobian of a Jacobian and hence can be 
expressed in terms of forms of lower degree, § 77. 

In fact, to give the actual expression, we have 
{(H,f), ^] = i (/, ^fH-\ (H, iryf, 
whatever "^/r may be, since (H, fy = 0. 
Therefore - (tj) = i (fjy H-\ (Hjyf, 

or (t,f) = -\H\ 

Next 

(t,fy=^-\(ha)hxai,bx'Y 

= - (ha) (hb) (ab) axbx + X [(hay ax, 6*^}, 
since (ha) (hb) (ab) a^ bx 

is one term in the transvectant. 



88-89] ELEMENTARY COMPLETE SYSTEMS 91 

Now this term vanishes, and since 

we have 

Finally 

(t, ff = - {{abf (ac) h o^\ d^f = - {ahj (ac) (bd) {cdf = - A, 
so that (/, tf = A an invariant which proves to be irreducible. 
The products formed from/I H, t which are of degree four are 
/, f^H, ft, H\ 

and all except H- may be rejected because t and / are both of 
order three. 

Further {H^,fy is reducible unless p > 2 because H- contains 
the factor H whose order is two. 

Hence the only possible irreducible form of degree five is {H'\ff. 
But 

{H\fy = {K'h'^\ a^J = {haf{h'a)h'^ 

= — [{hd)- ax, h'a;-] = since (ha)'- a^ = 0, 

hence there are no irreducible forms of degree five, and in fact it 
is easy to see that there are no more irreducible forms. For if 
there were, the one coming next in ascending degree would be of 
the form 

(f^HHyjy. 

The only products that can lead to irreducible forms are 
f, t, H and H-, because when /3 > 2, H^ involves the factor H- 
whose order is greater than three ; but the transvectants arising 
from each of these products have already been considered, hence 
there are no more irreducible forms ; in other words, every 
invariant or covariant of the cubic is a rational integral function 
of / H, t and A. 

89. Irreducible system for the quartic. If the form be 

J = Qjx =^ Ox = Ca- , 

then the irreducible forms of degree two are 

H = (abyax^hx- and i^{ab)*, 
H being of order four and i an invariant. 



92 THE ALGEBRA OF INVARIANTS [CH. V 

The only product of degree two that we need consider is H, 
and hence the irreducible forms of degree three are included in 

{Hjf, (Hjy, (Hjy, {Hjy. 

Now 

(S>fy = W (ac) a^bJ'Ca,^ = - t 
where t is an irreducible covariant. 

{H,fy={{abyaJ^K\c,^Y 
= j\ (aby (acy bjc^' + j\ (aby (ac) (be) a^b^c^^ + ^ (aby (bey a^W 
= i (aby (acy ba^'c^' + f (aby (ac) (be) a^b^c^^ 
= i (aby (acy b^^c^^ + \ (aby c^^ [(acy b^' + (bey a/ - (aby c^^] 
= (aby(acyb^'cJ'-Uahy.Ca?, 
since the symbols are all equivalent. 
Then since, § 22, 
(aby c^' + (bey a^' + (cay b^* 

= 2 (aby (oAiy ba?e^' + 2 (bay (bey a^^d + 2 (cay (cby a^b^, 
we have (aby (acy b^c^' = \ (abf . e/. 

Therefore (H,fy = W- W= ¥f 

and is reducible. 

Next (Hjy = {(aby aj'b^', e^'Y = («^)' i^c)' W b^ 

for the two terms in the traiisvectant are equivalent. By inter- 
changing a and c it follows that (H,fy = 0. 

Finally 

(H,fy = {(aby a A', Ca^'Y = io,by (bey (cay =j 

an invariant ; hence the only irreducible forms of degree three 
are t and j. 

The only product /"H^ty of degree three that we need 
consider is t, and the only possible irreducible forms of degree 
four are therefore 

(tjy, (tjy, (tjy, (tjy. 

As a matter of fact these are all reducible. 



89] ELEMENTARY COMPLETE SYSTEMS 93 

To calculate them we use the series of ^ 54, with the scheme 



f H f 

4 4 4 
r 1 



which gives 



or 



'4 — r\ /I 



/4 - rW 

=2^v^^i(/,/)-^)-'. 



If r = 1 we find 

since (f, Hy = ^if, 

hence (t,f) = ^H"-j\if'. 

Ifr=2, 

(t,fy + {{f,Hyj] + j'A(fHfJY 

and since (f,HY = ^if, {fHf = 0, 

this gives (^./)' = 0- 

From r = 3, 

(t, fy + f {(/> sy-jY + A {(/ ^)^/} + i (/ ^)^ •/ 

= Ufy,H} + ^(/,/y.H, 

or a/>'^-i^•5^ + lJ/=i^i^, 

on putting in the values for the transvectants off and H. 

Thus 

{t,fy=i{iH-jf). 

The series does not apply when r = 4 because then r + 1 > 4, 
but it is easy to calculate 

(tjy 

directly or as in § 94. 



94 THE ALGEBRA OF INVARIANTS [CH. V 

For 

{(ah) a^'h^\ K*Y = («^0 («&)' ^* V + ^ \{ahy a^%^\ b^'Y 

+ fji {(ahy a^K, h^*Y + ^ {i<^hy, b^'}, 
while (aby {ah) b^^h^^ = [{aby a^b^, V} = 0, 

{ahya^'h^' = iif; (/,/>' = 0, 
{ahy a^hx = 0, 
hence (^,/)* = 0. 

Hence there are no irreducible forms of degree four. In fact 
there are no more irreducible forms, because the next in order of 
degree would be of the form 

(/"HHy,/)". 

Now since /, H, t are each of order four at least, irreducible 
forms can only arise from products containing each of the three 
by itself, and all these, viz. 

ifjy, (sjy, (t,fy 

have been already considered. Hence every invariant or covariant 
of the quartic is a rational integral function of/, H, t, i and j. 

90. Quintic. To illustrate still further the method of this 
chapter we shall apply it to some extent to the binary quintic. 
The covariants of degree two are 

(/,//= ZT and {f,fy = i. 

The products of powers of /, H and i which are of degree two 
are /^ H and i, and to find the covariants of degree three we 
have to consider transvectants of these three forms with f. 

The transvectants arising from /-' may be neglected, and hence 
we are left with 

{H,f)\ (H,/)\ (Hjy, {H,f)\ (Hjy, 

{i,f)\ (i/y 

as the only possible irreducible covariants of degree three. 
Now of these 

(H,fy = {{aby a^b^\ c^'^Y 



89-90] ELEMENTARY COMPLETE SYSTEMS 95 

and involves a term ^ 

{aby (ac)' a^bai'c^', 

.-. (Hjy = (aby (acy a^b^^c^^ + \ {(aby a^b^, c^»|« 

= ^axKcx {{ciby (acy b^Cx' + (6af (6c)- c^^ai 

+ {cay (cby ax%^] + \if 
= ^a^hcx I {(aby Cx* + (bey a^' + (cay 6^^} + \if 
= (\ + ^)if, 
so that (H,fy is reducible. 

Again (H, fy = \(aby a^b^, c^ 

and contains the term 

(aby(acybxHx\ 

This terra can be transformed so as to contain (ocV and hence 
must be a multiple of 

(aby (ac) b^c^* 

since the letters are equivalent. 

.-. (H,fy = X (aby (ac) b^Cx* + fi {(aby a^bx, c^»| 

= (\ + fM)(i,f). 

Further 

(H, fy = (aby (bey (acy a^b^c^ + \ a fy 

and 

(H,fy = (aby(bcy(acya, 

= i (bey (cay (aby {(bo) a^ + (ca) b^ + (ab) c^] = 0. 
Finally (i,fy is an irreducible form and 
(i,fy = (aby(ac)(bc)c,^ 

= - i (be) (ca) (ab) {(aby c^' + (bey a^' + (cay b^'} 
= - (bey (cay (aby a^b^c^ (§ 22). 

Hence the only irreducible covariants of degree three are 

(H,f) = t,(f,i) 

and (/, iy = - (bey (cay (aby a^b^c^ = -j. 

The reader may now hnd the irreducible forms of degree four 
and verify the result by reference to the chapter on the quintic. It 
will be seen at once that the method leads to much labour, that 
the reduction processes are not easy to discover, ai^d, when we 
mention that for the quintic we have to proceed step by step 



96 THE ALGEBRA OF INVARIANTS [CH. V 

until we get to degree 18 before the irreducible system is 
obtained, the impracticability of these methods in dealing with 
forms of order greater than four will be at once admitted. 

91. Further Theory of the Cubic. Syzygy among 
the irreducible forms. There is an identical relation con- 
necting the irreducible concomitants of the binary cubic — the 
simplest example of what is known as a syzygy among the 
covariants of a single binary form. 

In fact since t is the Jacobian of/ and H we have, § 78, 

- 2t^ = (f,ffH^ + {H, Hyp-^iHJfHf. 
Now 

{H,ff=^{ahfa^ = 

(H, Hy = [{ahf a^h, {cdf cM' = W (c^)' («c) (bd) = A, 

for although there are four terms in the transvectant they are 
identical in value, and we have 

the relation required. 

92. Since every covariant of the cubic is a rational integi-al 
function of/, H, t and A it follows that all expressions derived by 
convolution from products of powers of/ JET and t can be expressed 
in terms of/ H, t and A. 

The form of the expression can be easily inferred by con- 
sideration of its degree and order, but the actual determination 
of the coefficients may be a troublesome process. 

As an example consider the Hessian of t, i.e. {t, tf. It is 
a covariant of degree six and order two since t is of degree three, 
and the only product of / H, t, A fulfilling these conditions 
is ^A. We at once see that (t, tf is a numerical multiple of ZTA. 

The reader may calculate the actual value directly by using 
the series of § 54 — we give an alternative process. 

Let J=(t,f), 

then - 2J^ = {t, typ + if, ff P - 2 (/ tyft 



90-93] ELEMENTARY COMPLETE SYSTEMS 97 

But {t,f) = ^H\ {f,fy=H, (f,ty=o 

and t' = -\H^-l Ap ; 

therefore -^H' = (t tfp ^H{-\E'-\ A/^) ; 

that is {t,tfp=\ApH 

or {t,ty=lAH. 

Again, consider the symbolical product 
{ahy (ac) (bd) (cd) Cxdx 
which represents a covariant of degree four and order two. 

Since there is no product 

of this degree and order the covariant in question must vanish 
identically. 

To verify this we remark that, on interchanging a, b and c, d, 
the expression 

(aby (ac) (bd) (cd) c^d^ 

changes sign ; hence it vanishes. 

Ex. (i). Calculate the following transvectants in terms of/, II, t, A, viz. 

{ff, H), {H, H)% it, t), {t, t)\ {t, tf, (IT, t), {H, tf. 

(The only one presenting any difficulty is {H, t) and this is the Jacobian 
of a Jacobian ; its value is — ^ A/.) 

Ex. (ii). Shew that any symbolical product involving the factor {aKf 
vanishes identically. 

Ex. (iii). Shew that 

{m,pf = 0, {H^fif = 0, {H\P)=^H'^f{HJ)=-H^ft. 

93. Further Theory of the Quartic. As in the case 
of the cubic, the square of the covariant t can be expressed 
rationally in terms of the remaining forms. 

In fact we have, § 78, 

- 2t^ = (f,ffH^-2(H,fyHf+(H, Hfp, 

while {f,fr = H, (HJf = iif, 

so that it only remains to calculate (H, H)'\ 

G. & Y. 7 



98 THE ALGEBRA OF INVARIANTS [CH. V 

Now using the series of | 54 with the scheme 

( 4, 4, 4 J 
\ 0, 2, 2 / 

we have 

(') (') (') (') 



'f7') 1'7'' 



or 

{H,Hr+[{f,f)\H]^^{fjy.H 

= {(/ ^)^/}^ + ((/, Hr,f\ + 1 (/ Hy.f, 

that is (£r, Hy + liH = ii (/ /)^ + ^ j/, 

since (/, H)'^ = etc. 

Hence {H, Hf = iJf-^iH. 

Consequently 

which is the syzygy required. 

94. We shall illustrate the reduction of covariants of the 
quartic by calculating the values of the transvectants of 

/, H, t, 
taken two together. 

The transvectants of f with itself, H and t have already been 
found. 

As regards the trans vectant 

(H, Hf 

we remark that it vanishes when r is odd and 

(H,Hy = yf-iiH. 

There only remains (H, Hy. 

This is equal to 
{(ahy a^'h\ h^*]* = (ahy {ahy (bhy = {{ahy a^'h^\ hJ'Y 

= [{H,fy,fY=ii{f,fy = li\ 



93-94] ELEMENTARY COMPLETE SYSTEMS 99 

To calculate the transvectants 

{t,Hr, r>>3, 

apply the series of § o4 with the scheme 

/ H, f, H^ 

( 4, 4, 4 ) 

\ 0, r, 1 J 

and we have 

?)(i),„ ..„,„,. .^r 






I 



On taking r=l, 2, 3 successively and putting in the values 
of the transvectants (H, HY'^^ we find 

(t,H)=if{iH-jf) 

{t, Hy = 

For {t, Hy, the scheme 

/ H, f, H^ 

4, 4, 4 
V 1, 3, 1 / 

must be used ; or else as in the case of (t,fy it is easy to see that 

{t,Hy = o. 

To find (t, ty we apply the series with the scheme 

{ 4, 4, 6 ) 

\ 0, 2, 1 / 
which gives 

{(/ H), tY + {{/, Hy, t] + j\ if, Hy t = u ty, h] + § (/ ty h 
or {t,ty + \{f,t) = l{f,ty.H. 

On substituting the values for {t,f) and {t,fy we find 

(t,ty = ujfff-^^H^'-^iT)' 

7—2 



100 THE ALGEBRA OF INVARIANTS [CH. V 

For {t, ty we use the scheme 

//' ^' «\ 
I 4, 4. 6 ) 

\ 1, 3, 1 / 

leading to 

{t,ty+l(f,ty={{t,fy,BY 

-\is,Hy-iif,Hy 

and hence {t, ty = 0. 

Finally 

(t, ty = {{t, fy, HY = i i (H, Hy - ij (/ Hy 

Ex. (i). Deduce the value of {t, tY from the relation 

-2{(«,/)}2=(«, typ+(f,fyt'-2(f, tfft. 

Ex. (ii). Apply Gordau's series to calculate {t, t)^ for the cubic. 
Ex. (iii). Prove that 

Ex. (iv). Prove that for the quartic 

{{ff,iif,H}=+yt 

{{H,H)\HY = \f-^z\ 

Ex. (v). Prove that the Hessian of the Hessian of the Hessian of a 
quartic / is 

Ex. (vi). Calculate the values of {{t, t)'^, ty for r=l, 2, 3, 4, 5, 6. 

Ex. (vii). If a quartic / be the product of a cubic by one of the linear 
factors of its Hessian, then 

(/,/)*=o. 



CHAPTEE VI. 

GORDAN'S THEOREM. 

95. We have already referred to Gordan's theorem which 
asserts the existence of a finite complete system of covariants 
for any binary form, and, in fact, we have illustrated the truth 
of the theorem in obtaining the complete systems for the 
quadratic, cubic, and quartic. Our previous method is of little 
practical utility in dealing with forms of order greater than four, 
but a comparison between it and the procedure of Gordan may 
not be without value as a primary indication of the salient 
features of the latter. In the last chapter covariants were 
classified according to their degree and we shewed how to obtain 
those of degree m by transvection from those of less degree. In 
Gordan's investigation covariants are classified according to their 
grade — the grade being a definite even number associated with 
any symbolical product, § 61, and all covariants of grade 2r are 
obtained by transvection from those of inferior grade together with 
some of grade 2r. 

The advantage of using the grade is that no covariant can be 
of grade greater than n ; accordingly, the number of steps in the 
process is small, whereas there being no limit, cb priori, to the 
degree of an irreducible covariant, and the actual degree reached 
by irreducible forms of quintics, etc., being very high, the number 
of steps in the other process is uncertain and at the best large. 
As will be seen later, on the other hand, the transition from grade 
2r — 2 to grade 2r is commonly much more difiicult than that from 
one degree to the next higher. 

Several preliminary propositions are necessary before we can 
undertake the actual proof of the existence of the complete 
system ; these we now proceed to explain. 



102 THE ALGEBRA OF INVARIANTS [CH. VI 

96. The first lemma required belongs to that branch of the 
theory of numbers known as Diophantine Equations. 

For the sake of clearness we shall begin by giving an 
illustration. 

Consider the homogeneous linear equation 

2x + 5y = 2z ; 

it is easy to see that the number of solutions in positive integers 
is infinite. 

Moreover, if 

and so=p\ y = q, z = r\ 

be two solutions, then 

x = 'p-\-'p, y = q + q', z = r + r 

is also a solution. 

We shall call this latter the sum of the former two solutions, 
and when any solution can be written as the sum of two smaller 
solutions (throughout we deal only with solutions in positive 
integers), it is said to be reducible. Otherwise a solution is 
irreducible, and the important fact for us is that the number of 
irreducible solutions is always finite. 

Thus for the equation above the only irreducible solutions are 

x = S, y = 0, z = 2; a; = 0, y = S, z = 5; 

a;=l, y = 2, 2^ = 4; x=2, y = l, z = S. 

In fact if a; > 3, then z>2, and the solution can be reduced 
by means of a; = 3, y = 0, ^ = 2 ; whereas if y > 3, then z>5, and 
the solution can be reduced by means of « = 0, 2/ = 3, ^= 5 ; thus 
in an irreducible solution neither x nor y can exceed 3, and, as 
the number of remaining possibilities is finite, the irreducible 
solutions can be easily found by trial. 

By continually reducing a given solution, say a; =p, y = q,z = r, 
we can express it in terms of the irreducible solutions, that is in 
the form 

p = SX + v + 2p ) 

q = Sfi + 2v + p i (A), 

r=2\ + 5fi + 4iv + Sp) 

where \, fj,, v, p are positive integers : 



r 



96-97] gordan's theorem 103 

e.g. take the solution 

x = oO, y = 7, z = 45, 

reducing by means of 

a; =3, 2/ = 0, z = % 

a;=16.3 + 2, 2/ = 7, ^=16.2 + 13; 
then reducing 

x' = 2, y' = 7, 0' = 13 

by means of a; = 0, 2/ = 3, 2^ = 5, 

a;' = 2, y = 2 . 3 + 1, / = 2 . 5 + 3, 

and the remaining part 

a^' = % 2/"=l. /' = 3 
is irreducible. 

Hence in this case we have 

\=16, fi=^1, v = 0, p = l. 

Of course if we substitute the expressions in (A) for x, y, z 
the equation is satisfied identically ; the important point is that 
evei^y positive integral solution can be written in the form there 
indicated. 

97. The idea of reducibility can be extended at once to 
any number of linear homogeneous equations, for the sum of two 
solutions is always a solution, and we may enunciate our first 
lemma as follows : 

The number of irreducible solutions in positive integers of a 
system of homogeneous linear equations is finite. 

Consider first a single equation, 

a^Xi + a.x^ + . . . + amXm = &i 3/1 + ^2^2 + . • • + hnyn, 

connecting the x'b and y^, where the coefficients a, h are positive 
integers. 

If the two solutions 

^i^^Si) '^2^62' ••• '^m~ ^m> yi^^Vi' y-2^V2t ••• yn^^Vn'} 
^1 = ^1, ^2 = ^2, . • • ^m = ^m, 2/1 = Vi, 1/2 = V2, • • . yn = Vn, 

be typified by 



104 THE ALGEBRA OF INVARIANTS [CH. VI 

respectively, then 

also typifies a solution and this latter is reducible. 
First the equation has mn solutions of the type 

with the rest of the variables zero. 

Next suppose that in a solution one of the xs (say x-^ is 
greater than 

61 + 62+ •■• +&«, 

then the right-hand side of the equation must be greater than 

ai(&i + &2+-.. + 6n), 

i.e. 61 (2/1 - Oi) + 62 (2/2 - ai) + . • . + hn (y« - aO > 0, 

so that at least one y must be greater than a^. Let yr>(h, then 
the solution in question is reducible by means of the solution 

with the other variables zero. 

Hence (61 + 62 + . . • + 6„) is an upper limit to the value of any 
X in an irreducible solution,. similarly (oi + ^2 + • • • + ^m) is an upper 
limit to the value of any y ; but the number of solutions reducible 
or irreducible subject to these restrictions is manifestly finite, 
therefore d fortiori the number of irreducible solutions is finite. 

If the irreducible solutions be typified by 

«? = «!, 2/ = /3i ; 
sc = (h, y = ^2] 

« = «P' y = ^i» 

then by continued reduction any solution can be expressed in 
the typical form 

a; = ^iffi + ^2«2 + . • • + t^ap 

y = t,^, + t^^,+ ...+t,/3„ 

where the t's are all positive integers. 

Suppose now we have a second equation of the same nature 
between the variables ; then on replacing the x's and y's by their 



97] gordan's theorem 105 

values in terms of the ^'s the first equation will be satisfied 
identically and the second equation will become a linear equation 
between the ^'s with integral coefficients. 

Hence by the above reasoning every solution of the equation 
among the ^'s may be written in the typical form 

^=T,7i+ 2^272 + ... +r,7<., 
where t = yi, ^ = 72, ••• t = ya^ 

typify the irreducible solutions, and the T's are all positive 
integers. 

Now substitute these values for the t's in the expressions for 
the xs and y's and we find at once that 

where the /c's and X,'s are fixed positive integers. 

Thus the only possible irreducible solutions of the two equations 
are those typified by 

X = K^, y ^= \,i\ X =^ K2, y ^^ A<2 J ... X ^ K(r , y ^ A,(, , 

for every other solution can be expressed as a linear combination 
of these. 

If we had a third equation, on substituting for the xs and y's 
their values in terms of the T's the first two equations would be 
satisfied identically, and the third would become a linear equation 
among the T's. Then this equation in turn has only a finite 
number of irreducible solutions, and hence, reasoning exactly as 
before, we should find that the three equations given have only 
a finite number of irreducible solutions. The process can be 
manifestly extended to any number of equations, and hence our 
theorem is established. A formal proof by induction from (r — 1) 
equations to r equations could of course be easily given. 

Ex. To find the irreducible solutions of the two equations 

x-\-w=y + z)' 

The irreducible solutions of the second equation are easily found since no 
letter can exceed 2 ; they are 

x=\, y = l; x = \, 2 = 1; y = l, w=\; z=l, w = l; 

variables not naentioned in a solution being zero. 



106 THE ALGEBRA OF INVARIANTS [CH. VI 

Hence the general solution of the second equation is 
a;=a + b, i/ = a + c, z=b+d, w = c+a, 
where a, 6, c, d are positive integers. 
The first equation now becomes 

and for an irreducible solution a, 6, c cannot exceed 2. 
On trial we find the following irreducible solution 

a = 2, c?=5; 6 = 2, d=\; c = l, d=\; a = l, 6 = 1, d=Z. 
Hence the general solution is 

a=2a + S, 6 = 2/3 + S, C=y, c?=5a+/3 + y + 38. 
These values for a, 6, c, d give 

a7 = 2a + 2i3 +2S\ 

y = 2a + y + 

« = 5a + 3j3+ y + 4S 
20 = 50+ /3 + 2y + 3S, 
as the general solution of the two equations. 

The only possible irreducible solutions are accordingly 
^=2, y = % z=5, w=5 
x=2, y=0, 2 = 3, w=l 
x=0, y=l, 2=1, w = 2\ 
x=2, y = \, 2=4, xo=Z} 

Of these the first is the sum of the third and fourth, while the foui-th is 
the sum of the second and third, so the only irreducible solutions are the 
second and third. In other words any solution of the two equations may 
be written in the form 

x=2p, y = q, z = 3p + q, w=p + 2q, 

where p, q are positive integers. 

Ex. (i). Prove that the equation lx + 4y—3z has four irreducible solutions 
and that every solution of the two equations 7x + 4y = 3z, z+5w=2y can be 
written in the form 

x=2a + c, y = 7a+l5b + Uc, 2 = 14a + 206 + 17c, w = 2b + c. 
Ex. (ii). Find the number of irreducible solutions of the equation 

11 •" 2 2 ■" • • • "t" n n ^~ ^3C^ 

the a's being positive integers. 

Ans. If all the a's are even there are n irreducible solutions, if r of 

rir—V) 
the a's are odd there are « + — solutions. In an irreducible solution 

at the most only two of the letters on the left are different from zero. 



97-99] gordan's theorem 107 

Ex. (iii). Prove directly that in an irreducible solution of the two 
equations 

hyi+hy2+ ••• +Kyn=y+z 

X is less than the greatest a, and y is less than the greatest h. 

98. System of forms derived by transvection from two 
given systems. Consider two systems of binary' forms in the 
same variables a^i, x.^, viz. 

J-i, A.2, ... Am, of orders a,, a^, ... a^ respectively 

and Bi, B^, ... B^, of orders h^, b^, ... bn respectively; 

we suppose each form written symbolically and denote by U, V 
two products of the types 

A,-^A,-^ . . . Am'^'n, B/^Bl^ . . . B,fn 

wherein all the exponents are either zero or positive integers. 

The system C is said to be derived from the systems A and B 
by transvection when it includes all terms in all transvectants of 
the form 

{U, V)y. 

It is clear that some of the members of the system C are 
reducible, that is they can be expressed as rational integral 
functions of simpler meijibers of that system — in fact, if 

then there are many terms in the transvectant 

{U, V)y 
which are products of two terms, one belonging to the transvectant 

and the other to the transvectant 

(U,, V,)y^. 

99. We can now enunciate and prove our first theorem, viz. 

The number of transvectants of the form ( U, V)y which do not 
contain reducible terms is finite. 

* It is of course assumed that y^, y^ are such that these transvectants 
are possible, e.g. y^ must not exceed the order of V-^. 



108 THE ALGEBRA OF INVARIANTS [CH. VI 

For suppose that any term of the transvectant 

contains p symbols of the A's, not in combination with a symbol 
of the B's and a symbols of the B'& not in combination with a 
symbol of the J.'s, then we have 

because each side of the first equation, for example, represents the 
total number of the symbols of the forms A which occur in the 
product U. 

Now to each positive integral solution of the above equations 
in a, y8, p, o", 7 there correspond definite products U, V and a 
definite value of 7 and hence a unique transvectant. But as we 
have already remarked if the solution corresponding to (17, V)y 
be the sum of those corresponding to (U^, Fi)^' and {17.2, ^■2)'^^ 
then (U, V)y certainly contains reducible terms. Hence trans- 
vectants corresponding to reducible solutions always contain 
reducible terms and inasmuch as the number of irreducible 
solutions has been proved to be finite it follows that the number 
of transvectants not containing reducible terms is finite. 

100. In actually finding the transvectants which do not 
contain reducible terms we may use the equations (I), but it is 
generally easier to proceed directly. 

Suppose that the system A contains the single form /= a^^ 
and the system B the single form i = hi, then we have to consider 
transvectants 

If 7 > 2/3 this transvectant vanishes, if 7 < 2y3 — 1 it contains 
terms of i (/", i^~^)t ; hence for an irreducible transvectant we 
must have 7 = 2/3—1 or 2/3, In the same way 

7 :^ 5a and -^ oa — 4. 

Again if a > 2, then for an irreducible transvectant 7 > 10 
and hence /3 > 4, so that some terms may be reduced by means 
of (/^ i^Y. Thus we need only consider a = 0, a = 1 and a = 2. 

For a = we have i. 

For a=l we have/ (/i), {f,i)\ {f,i-)\ {f,ij, {f,i^)\ 



99-101] gordan's theorem 109 

For a = 2 we have {f\ i'Y, (/^ iy, {f\ i*y, (/^ i% 

Now {f^, iy contains terms which are products of a term 
of (/, i'^ and (/, i)'^ and a like argument applies to 

(/^ i% (A i% (A i'f> 

so the only transvectants not containing reducible terms are 

/ i if, i), if, if, (/ i% (/, i')\ (f, iJ. (A> iT- 
Ex. (i). If/ be any form of order 2n + l, then the transvectants 

which do not contain reducible terms are 2?i + 4 in number. 

Ex. (ii). Find the corresponding result when / is a form of even order. 
Ex. (iii). The only transvectants 

where /[=aa;* and/2=6a;^, which do not contain reducible terms are 

/i, /2, h (/i, i\ ifi, ^7, (/i> i'r> (/i' ^')s 

(A, i), (/2> if, (A, i')\ if2% i'f- 

101, Definition. The system of forms A is said to be 
complete when any expression derived by convolution from a 
product U of powers of the forms A is itself a rational integral 
function of the A's. 

Thus for example the system of forms 

H = (abfa^bx 

t — {ahy {ca) hxC^ 

A = {aby(cdy{ac){bd) 

is complete because any expression derived in the above manner 
is a covariant of/ and therefore a rational integral function of 
/, H, t and A. Again the system H and A included in the above 
is itself complete. 

More generally the system A is said to be relatively complete 
for the modulus G consisting of a number of symbolical deter- 
minants when any expression derived by convolution from a 
product i7 is a rational integral function of the ^'s together 
with terms involving the modulus G. 



110 THE ALGEBRA OF INVARIANTS [CH. VI 

Thus the system consisting of a single form 

is relatively complete for the modulus (aby, since any expression 
derived by convolution from a power of f can be transformed so 
that a factor (aby occurs in it. 

Again for a quartic the system 

/= a^^ H = (aby aA^ t = (aby (ca) aj)^^c^\ 

j = {bey {cay {aby 

is relatively complete for the modulus {aby, for all covariants of / 
are rational integral functions of 

f,H, t,j,i, 
where i = {aby. 

We may extend our definition of relative completeness still 
further : a system A is said to be complete relatively for several 
moduli Gi, 0^ ... when any expression derived by convolution 
from a product C is a rational integral function of the A's together 
with terms involving one at least of the moduli G^, G^ 

It will be seen later (or it can be verified without difficulty) 
that in connection with any quantic a^^ = bx^ ... the single form 

H = {aby a^^'-^b^''-'' 

is relatively complete with respect to the modulus {aby except 
when n = 4. 

If n = 4 the complete system worked out for the form H shews 
that any expression derived by convolution firom a power of H is 
a rational integral function of H together with terms involving 
i or j. 

That is H is relatively complete for the two moduli {aby and 
{bey {cay {aby. 

It will be noticed that a complete system is relatively complete 
for any modulus or systems of moduli. 

102. The system G derived by transvection from two given 
systems contains an infinite number of forms, but it is said to be a 
finite system when all its members can be expressed as rational 
integral functions of a certain finite number of them. More 
generally it is said to be relatively finite for a given modulus G 



101-103] gordan's theorem 111 

when every member of C can be expressed as a rational integi"al 
function of a certain finite number of them together with terms 
all of which involve the modulus G. 

For example the number of covariants of a binary cubic is 
infinite but inasmuch as every one is a rational integral function 
of/, H, t and A the system of forms is said to be finite. 

Again it will be seen later that every covariant of the binary 
w-ic 

can be expressed in terms of/, H, t, where 

t = {ahf (ca) a^-^h^'-^c^-"- 

together with terms involving the factor {ahf. We should state 
this fact thus — The system of covariants of a binary n-\Q is 
relatively finite for the modulus {ahf. 

103. Theorem. If the systems of forms A and B are both 
finite and complete, then the system derived from them by transvection 
is finite and complete. 

(a) The system is finite. 

In the proof of this theorem we shall consider the transvectants 

{U, V)y 
in a certain order defined as follows : — 

(i) Transvectants are taken in order of ascending total degree 
of the product UV in the coefficients of the forms involved in A 
and B. 

(ii) Those for which the total degree is the same are taken in 
ascending order of indices. 

Further than this the order is immaterial. 

With this convention let T and T' be any two terms of the 
transvectant 

{U, V)y, 
then {T-T') = L{U,T)y' 

where 7' < 7 and U, V are derived by convolution from U, V 
respectively. 



112 THE ALGEBRA OF INVARIANTS [CH. VI 

But since the systems A and B are complete 

U = F{A\ 

where F{A) is a rational integral function of the J.'s, that is, an 
aggregate of products of the type TJ, and ^ (jB) a similar function 
of the 5's. 

Thus {iJ^yy 

can be expressed as the sum of a number of transvectants in each 
of which the index is less than 7. By hypothesis all such 
transvectants have been examined before the one now under 
consideration and hence if all the (7's derived from previously 
considered transvectants can be expressed in terms of 

Cj, Cj, ... Cy, 

then all C's up to and including those derived from 

can be expressed in terms of 

Ci, C^a, ... Cry T, 

where T is any term of the last transvectant. 
But if the transvectant 

contain a reducible term, say T=T^T^, then inasmuch as T^, T^ 
must both arise from transvectants previously considered no term 
T need be added to the system 

Cj, C2, ... Gf. 

Thus in gradually building up a system of C's in terms of 
which all O's can be expressed we need only add a new member 
when we come to a transvectant containing no reducible term and 
then we need add only one new member. But the number of 
transvectants containing no reducible term is finite and hence a 
finite number of (7's can be chosen such that every other is a 
rational integral function of these, that is the system G is finite. 

Remark. A set of C's in terms of which all others can be 
expressed rationally and integrally can be chosen in various ways, 
for any term may be selected from each transvectant containing 
no reducible terms. Further since the difference of two terms of a 



103] gordan's theorem 113 

transvectant can be expressed by means of terms of transvectants 
previously considered we may, instead of choosing a single term 
from any transvectant, take an aggregate of any number of such 
terms or even the transvectant itself, and it will still be true that 
every member of G can be expressed as a rational integral function 
of the members of our finite system. 

(6) The finite system so constructed is complete. 

i-iet Oj, L/2, ••' 0,. 

be the finite system, then we have to prove that an expression W 
derived by convolution from any product of the form 

is a rational integral function of C^, C^, ... C^. 

Suppose that W contains p determinantal factors in which 
a symbol belonging to a form A occurs in combination with a 
symbol belonging to a form B. 

Then W is a term in a transvectant 

(u, vy, 

where U contains only symbols of the A's and V only symbols of 
the B's, so that U is derived by convolution from a product U of 
the ^'s and V is derived by convolution from a product V of 
the B's. 

Thus W={u, vy + tiuvy, 

where p < p and UV are derived from U, V by convolution and 
therefore ultimately from U, V. 

Now U=F(A), 

V=^{B), 

accordingly W can be expressed as an aggregate of transvectants 
of the form 

(U, vy. 

But we have just proved that every term of such a transvectant 
is a rational integral function of the C's and consequently W is 
also a rational integral function of them. 

Hence the system is not only finite but complete. 

G, & Y. 8 



114 THE ALGEBRA OF INVARIANTS [CH. VI 

104. Theorem. If a finite system of forms A, all the 
members of which are covariants of a binary form f include f and 
be relatively complete for the modulus H ; if further, a finite 
system B be relatively complete for the modulus G, and include one 
form Bi whose only determinantal factors are H, then the system 
C derived by transvection from A and B is relatively finite and 
complete for the modulus G. 

As an example of the theorem let A consist of 

f=cix= bx\ 
and B of the two forms 

E = {aby aj)^, A = {abf (ac) {bd) {cd)\ 

Then A is relatively complete for the modulus (a6)^ § 88, and B 
is absolutely complete, being the complete system of the Hessian 
of the cubic ; hence according to the theorem the system derived 
by transvection should be absolutely complete. This is obviously 
true, for the new system contains /, H, t. A, where 

t = {fH) = -{ohy{ac)b^c^\ 

and every possible member of the derived system is a covariant of 
/, therefore they are all rational integral functions of f H, t. A, 
which constitute the complete system of the cubic. 

105. Lemma. If P be derived by convolution from a power 
off any term in the transvectant 

(P, vy 

can be expressed as an aggregate of transvectants of the type 

{U, vy 

in which the degree of U is at most equal to that of P. 

(Throughout we shall use U, V as typical symbols for products 
of powers of the forms of A and B respectively.) 

This statement is manifestly true when the degree of P is 
zero ; assuming it true when the degree of P is less than r we 
shall establish it when the degree is r. 

In fact if T be a term in 

(P, vy, 
T={P,vy+X{P, vy- 



104-105] gordan's theorem 115 

and since P, P are derived by convolution from a power of the 
form f which is contained in A, 

P=^F{A) + HW*, 

P = F'{A) + HW', 

while V=^{B) + GZ=^{B), mod a 

Hence T can be expressed as the sum of three parts ; 

(i) transvectants of the type [F{A), ^{B)Y the degree of 
F{A) being r ; 

(ii) transvectants of the type {Q, Vy, where Q is of the same 
degree as P and contains the factor H ; 

(iii) terms containing the factor 0. 

Now Q can be derived by convolution from 

where s is less than r the degree of P ; therefore any term in 

can be derived by convolution from 

and is expressible in the form 

S(P',5rr), 

where P' is derived by convolution from /* and is of degree less 
than P. But by hypothesis every terra in these transvectants 
can be expressed as an aggregate 

t{U, vy, modG^, 
for 57F=^(P), modG^, 

where the degree of U is at most equal to s and therefore less 
than r. 

On referring to the expression for T we see that T can be 
written in the form 

^{U, vy, modG; 

consequently the statement in the lemma can be completely 
established by induction. 

* HW simply means a symbolical product containing the factor H. 

8—2 



116 THE ALGEBRA OF INVARIANTS [CH. VI 

Cor. If the product P contain the factor H, then any term in 

(P, Vy 

can he expressed in the form 

where the degree of U is less than that of P. 

For P is now of the fonn Q just discussed, and any term in a 
transvectant 

(Q, vy 

can be expressed as a sum 

t(U,vy 

in which the degree of U is at most equal to s which is less than 
the degree of P. 

106. The proof of the theorem is now the same in principle 
as that in § 103. 

The transvectants are considered in the following order. 

(i) In order of ascending degree of UV in the coefficients 
of/ 

(ii) Those for which the degree of UV is the same are taken 
in order of ascending degree of U. 

(iii) Transvectants for which these two degrees are the same 
are taken in order of ascending index. 

Further than this the order is immaterial. 
If T and T' be two terms in 

{U, vy^ _ 

then T'-T=t{U, Vy\ 

where v <v. 

But U=F{A) + liW, 

therefore 

T'-T=t[F{A), ^{B)Y + ^[HW, <^{B)Y, modG. 
Transvectants of the type 

{FiA),^{B)y 
have been previously considered, for the degree of F(A) is the 
same as that of U and v <v; further by the lemma transvectants 
of the type 

[HW^^iB)]"' 



105-106] gordan's theorem ' 117 

can be expressed in the form 

where the degree of U' is less than that of HW, i.e. less than that 
of U. . 

Thus T'-T can be written 

2 ( U", Vy + t(U', Vy, mod G, 

where the degree of U" is the same as that of U and v' < v, while 
the degree of U' is less than that of U. 

Hence if all terms of transvectants considered previously to 

{u, vy 

can be expressed rationally and integrally in terms of 

Gi, C2, ... Gp 

(except for terms involving G); then all terms of transvectants 
up to and including 

{U, vy 

can be expressed in the form 

F{C„G,,... Gp,n modG, 
where T is any term of the last transvectant. 
If the transvectant 

(u, vy 

contain a reducible term we may suppose it to be T, and since 
T—T^T^ where T^, T^ are terms of former transvectants, there is 
no need to add the term T to Cj, Cg, ... Cp. 

It follows that in constructing a system of (7's in terms of 
which all (7s can be expressed we have to add a new member 
only when we come to a transvectant containing no reducible 
terms and then one only. The number of transvectants con- 
taining no irreducible terms is finite, § 99 ; hence if G^,G^, ... Gg 
be a series of terms one from each of this finite number of 
transvectants, any other member of the system G derived by 
transvection from A and B can be expressed as a rational integral 
function of G^G^, ... Gq together with terms involving the factor G; 
in other words, the system G is relatively finite for the modulus G. 

Next the system 

(7i, G^, ... Gq 

is relatively complete with respect to the modulus G. 



118 THE ALGEBRA OF INVARIANTS [CH. VI 

For any term T derived by convolution from 
may be regarded as a term in a transvectant 

(u, vy, 

where U is derived by convolution from a product of the A's and 
V from a product of the B's. 

Hence T can be expressed as an aggregate of transvectants 

{U,vy- 

while U=P can be derived by convolution from a power of / and 

V=^{B), modG; 
therefore T^^\P,<^ {B)Y, mod G, 

= X (P, vy, mod G, 
= t(U, vy, mod G. (Lemma.) 
Consequently, as has just been proved, 

T = F{G„G„... Gq), modG, 
and the system is complete. 

107. Cor. I. If the system B is absolutely complete, then 
the system derived by transvection from A and B is absolutely 
complete. 

Cor. II. If the system B is complete for two moduli G and G' 
and contains a form whose only determinantal factors are H, then 
the derived system is complete for the two moduli G and G'. 

To prove this we have only to write 

B = F{B), modd {G,G') 

instead of B = F(B), mod (G) 

at every stage of the foregoing proof. 

108. Gordan's Theorem. These long preliminary ex- 
planations are now at an end and the actual proof of the theorem 
does not present much difficulty. 

Every co variant of a binary form 

/=a^« = 6^" = etc. 

is either a power of / or else contains a factor (aby, and hence the 
form f itself is a complete system with respect to the modulus (aby. 



lOG-109] gordan's theorem 119 

Assuming now that a system of co variants containing / and 
relatively complete for the modulus (ab)^ can be found we shall 
shew how to construct a system also containing f and relatively 
complete for the modulus (a6)^+^ The system relatively complete 
mod (aby'' is called -4^-1, and since every covariant can be derived 
from / by convolution it is a rational integral function of the 
forms in A^^i except for terms involving the factor {obY'. 

To construct the system Aj^ when A^^^ is known we make use 
of the theorem of § 104. 

We must therefore begin by constructing a system 5fc_i possess- 
ing the following properties : 

(i) it contains the form {ahf^ a^^'-'^h^''-^, 

(ii) it is relatively complete for the modulus (a6)^+^ 

Then the system derived from ^^-i and Bj^_i by transvection will 
be finite and complete with respect to the modulus (a6)^+^ and as 
it obviously contains f which is contained in ^^-i it is the system 
A^ required. 

109. Accordingly we have now to shew how to construct the 
system Bj^^. . 

There are three cases. 

?i 

I. If 2A?<- then any. form derived by convolution from a 

power of H^. = {ab)'^ a^^'-'^hx'^-^'' is of grade {1k+l) at least and 
therefore of grade {2k + 2) since all symbols are now equi- 
valent. 

Hence Hj^ is itself relatively complete for the modulus (a6)^+'^ 
and in this case th« system B^ consists of the single form 

{ahf' a^-'^h^-'^. (§ 74.) 

II. If 2A; > ^ then H^ = (ab)* a^^^^^h^'^-^ is of order less than 

n, say m. 

Now we suppose that the complete system of co variants for 
a form of order < w is known and we derive a system from Hk 
on the model of the complete system of a^"* as explained in 
i 79, 80. 



120 THE ALGEBRA OF INVARIANTS [CH. VI 

Neglecting terms containing (a6)'*+2 ^q gan replace each copied 

form by a single term ; the system so derived is complete for the 

modulus (a6)^+^ and is therefore the system Bj^^i required. 

ft 
III. If 2k = ^ — a case which can only arise when w is a 

multiple of 4 — we have a rather different state of things. 

Here the form Hj^ = {ah)*ax^~^hx^~-^ is relatively complete 
for the two moduli 

(a6)^+S {ahY' {hc)"^ {caf\ 

the latter being an invariant J, and hence by Cor. II. § 107 the 
system derived by trans vection from A]c_y^ and ^j._i is relatively 
complete for the moduli (a6)^+^ and J\ calling this system G^ 
for a moment we have 

G, = F(C,) + J.P„ mod (abr^^ 
where Pi is a covariant of degree less than G^.. 

Further since Pj can be derived by convolution from / which 
is contained in G^, we have 

P, = F, (Cfc) + /. P^, mod {ahf+\ 

where Pa is a covariant of degree less than Pj. 

Proceeding in this way we see that Gk is a rational integral 
function of / and the forms in Gk together with terms involving 
the factor {ah)'*^^ 

Hence if we add J to the system G^ and call the total system 
Ajc it follows at once that A^ is relatively complete for the modulus 

Therefore in every case, given the complete system mod (a6)^ 
we can construct that mod(a6)'*+^; but the system A^ is/, thence 
we find the system A^, then from that the system A^ and so on, in 
fact we can construct the system A^ relatively complete for the 
modulus (a6)^+='. 

110. Consider now a little more closely what happens when 
we come to the end of the sequence of moduli {aby, {oh)*, {aby ..., 
and first let n be even and equal to Ig. 

Then the system Ag^^ is relatively complete for the modulus 
{aby^, and the system Bg^i consists of the single invariant (ab)^ so 
that it is of course absolutely complete. 



100-111] gordan's theorem 121 

Hence the system derived from Ag^i and Bg_i by transvection 
is absolutely complete and it contains/, therefore it is the complete 
system of invariants and covariants ; further since Bg^i consists of 
a single invariant the complete system Ag consists of Ag_i and 
that invariant {aby^. 

Secondly let n be odd and equal to 2g + l, then the system Ag_j^ 
can be constructed and it both contains /and is relatively complete 
for the modulus {ahy^. 

The system J5^_i is derived from the quadratic 

{aby^ a^bx 

by the same convolutions as the complete system of the quadratic 
oi^ = ^^ is found from this form. This complete system being a^' 
and {ci^y the system Bg^^ consists of 

{abyo aj}^, {abya {ac) (bd) (cdy^. 

This system is relatively complete for the modulus {ahy^'^^ 
by § 109 II, and this being a vanishing invariant it follows that 
jB^_i is absolutely complete. 

Hence the system derived from Ag_^ and Bg^^ contains / and is 
absolutely complete, that is it constitutes the complete system of/. 

To recapitulate — the complete system mod {aby can be written 
down at once, then from that we deduce the complete system 
mod {ahy and proceeding step by step we can finally construct 
an absolutely complete system as the last step in our series. 

We have therefore proved that the complete system is finite, 
for all the systems A^, A^, ... are finite, and we have shewn how 
to construct it on the assumption that the systems for forms of 
lower orders are known — the proof is thus inductive in its 
nature. 

111. We shall illustrate the above process by applying it 
to the quadratic, cubic, and quartic. 

(i) Quadratic. The system Aq is 

and the system B^ is {aby, hence the complete system is 

a^\ {aby. 



122 THE ALGEBRA OF INVARIANTS [CH. VI 

(ii) Cubic. Here Aq is 

fz=.ai = hi = etc. 
and 5i is (abf a^h^, {aVf {ac) (bd) {cdf, 

in fact H, {H,H)\ 

This system B^ is absolutely complete, therefore the system 
derived by transvection is the complete system. 

It consists of 

f,H,(H,Hy = A and {f'^,m)y. 

Proceeding as in § 88 we can shew that the only irreducible 
transvectant is (/, H). 

(iii) Quartic. Here A^ is 

/=««' = &«;'.•., 
£o is H = (aby a^bi, 

and this is complete modd (aby and (aby (bey (cay. 

The system derived by transvection is 

If 7 > 2 this has a term containing the factor (aby (acy which 
is congruent to zero modd (aby, (aby (bey (cay. 

Hence we need take only 7 = 1 and thence only a = l, /3 = 1, 
and we find that 

/ H. (f, H) 
is relatively complete modd (aby, (aby (bey (cay. 

Therefore / H, (f, H), (aby (bey (cay 

is complete mod (aby and is the system Ai. 

Then Bi being the invariant i = (aby we have for the 
complete system 

f,H,t = (f, H), i = (aby, j = (aby (bey (cay. 
112. We shall now apply the principles of §§ 73, 76 to the 

n 

deduction of a complete system mod (aby for the binary form of 
order n. 

The system ^o consists of 

/= a,;" = fta;" = etc. 

and £o of H = (aby a^-^ fca,"-^ .... 



111-112] gordan's theorem 123 

The system Ai is derived by transvection from Aq and Bq. 

Now (/-, H^)y 

has a term containing the factor {aVf{acf if 7 > 1, and since such 

a term is 

= mod {ahy 

the transvectant may be rejected. 

If 7 = 1 the transvectant contains reducible terms unless 
a = yS = 1, and hence A-^ consists of 

f,H,{f,H) = t 
The system Bi is 

(ahy a^-^ h^-^ 

and -4.2 is derived by transvection from A^ and B^. 

If the index of a transvectant be greater than two it contains 
a term having a factor {ahf {acf and this is 

= 0mod(a6)«. (§70.) 

We need only consider the cases in which the index is ^ 2, and 
since the order of each form in A^ is certainly greater than 2 (in 

fact ^ ^ 4), products of forms may be rejected. 

There remain transvectants of each form of Ai taken simply 
with 

H^ = {ahYa^''-'h^'^. 

For the future we shall only write down the determinantal 
factors of a covariant. » 

Transvectants with / give rise to 

{ahy {he), {ahy {hey. 
Those with H give 

{ahy {he) {cdy, {ahy {hey {cdy, 

and finally those with t give 

{ahy {be) {edy {dc), {ahy {bey (cdy (dc). 

Now by § 76 

(ahy (bey (cdy = (ahy (cdy, mod (ahy ; 

hence (ahy (hey (cdy (de) 

being a term of {{aby (bey (edy, e^"} 



124 THE ALGEBRA OF INVARIANTS [CH. VI 

we have 

(aby (bcf (cdy (de) = {(aby {cdy, e^% mod {aby, 

for all expressions derived by convolution from 

{aby {bey {cdy 

are ' =0 mod {aby. (§ 73.) 

Now a term of the last trans vectant is 

{cdy . {aby {ae), 

.-. {aby {bey {cdy {de) = {cdy . {aby {ae), mod {aby 

and accordingly may be rejected. 

Finally {aby {be) {cdy {de) 

is reducible as being the Jacobian of a Jacobian, and the system A^ 

consists of 

f, {aby, {aby {be), 

{aby, {aby {be), {aby {bey, {aby (be) {cdy. 

113. Before proceeding further we shall develope the results 
of § 76 by shewing that a symbolical product T containing the 
factor 

(a6)^ {bey {cdy, 

in which X is even and equal to fi + v, can in general be expressed 
in terms of covariants that are either reducible or of grade greater 
than X. 

The above reduction of 

^{aby{bey{cdy{de) 
is a case in point. 

In fact r is a term of 

{{aby {be)'' {cdy, 4>Y 

which we write {T, jty. 

Hence V = {T, j>y -^X{f,^y, p<p 

= {T, 4>y + 2 {T, 4>y', mod (a6)^+\ 

since T derived by convolution from {ahY {bey {caf is of grade 
greater than \, § 73. 

Again 

T={aby.{cdy-\-C^^, (§76), 

r = [{aby . {cdy, <f>]f> + {{aby . {cd)\ ^j"' + Cx+x. 



112-114] GORDANS THEOREM 125 

Now if 2n — 2X^p each of these transvectants contains terms 
having {cdy Cx"^'^ dx^~^ as a factor. 

Hence 

{{ahY.{cdY,^Y 
= (cdy {{ahf, <t>Y + 2 {{aby . (cdf, 0}', mod (abY+\ a < p, 

and by continuation of this process we can express T entirely in 
terms of reducible covariants and covariants of grade greater 
than X; it suffices to remark that the index a diminishes at 
every step. 

It is quite easy to see that the condition 

2n-2\^p 

is satisfied in all our cases — at any rate it will be in the course 
of the subsequent work. 

114. Returning now to the general form, B2 consists of 

Hs^iabyaaT-'ba;''-', 

and A3 is derived by transvection from A2 and B2. 

The argument used in evolving A^ and A.2 enables us to see 

(i) that transvectants with index > 3 may be rejected, 

(ii) thence that transvectants of products or powers of forms 
may be likewise rejected. 

We are therefore left with transvectants of the forms of -4 2 
taken simply with H^, the index being :}> 3. 

Omitting Jacobians of Jacobians and forms having a factor 

{aby (bey (cdy, where /j, + v^6, 
we have 

from/ (aby (be), (aby (bey, (aby (bey; 

H, (aby (be) (cdy, (aby (bey (cdy, (aby (bey (cdy; 

(aby (be), (aby (bey (cdy (de), (aby (bey (cdy (de) \ 

(aby, (aby (be) (cdy; 

(aby (be), none ; 

(aby (bey, {aby (be) (cdy (dey ; 

(aby (be) (cdy, none. 



126 THE ALGEBRA OF INVARIANTS [CH. VI 

Hence we have found for A3 the above ten forms in addition 
to those of A2. Putting aside the question as to whether any of 
these ten new forms are reducible, a continued repetition of the 
above process establishes the fact that all the forms of the 
system Aj^, relatively complete for the modulus 

are included in the set 

(abf {her (cdf {dey {eff" {fgY" ...» 

where the exponents satisfy the followipg conditions : 

(i) ^i^^h 

(ii) \, \', \", . . . are all even, 

(iii) \>X' + /M, V > \" + fjf, . . ., 

(iv) no two of the exponents /*, fi', ... are equal to unity. 

In fact 

(ii) follows immediately from the way in which the covariants 
are formed. 

(iii) results from the application of §§ 73, 76. 

(iv) is the expression of the fact that the Jacobian of a 
Jacobian is reducible. 

Ex. (i). If the orders to, n, p of the forms /, (f), yl^, be each greater 
than two, then 

Ex. (ii). For a form whose order is greater than four the covariants 
{abf (be) {cd), {ab) (bcf (cd), {ahf {bcf {cd) 
all vanish identically. 

Ex. (iii). If « > 5, then 
{bcf {caf {ahf aj"-* bx^-^c^"-* 

= (ab)* {acf a«»-6 Jx""* Cx"-2 - \ {abf a^^-s bj'-^ . Cx\ 

Ex. (iv). If » > 4, then 
and if w > 5, 

* Cf. Jordan, Liouville's Journal, 1876, 1879. 



114] gordan's theorem 127 

Ex. (v). For a form whose order is greater than three prove that 

Hence replacing (/, /)* by i express (H, 3)^ as a linear combination of 

Si, fii, /)',/'{/,/)' 
and finally express t^ in terms of the irreducible forms of the system. 

Ex. (vi). Prove that all irreducible covariants of degree four and rank 

not greater than - are included in 

{abf^ (bcf {cdf 

71 

where 2X :^ - and X > /i > v. 

Ex. (vii). In § 103 if no A be of order greater than m and no B be of 
order greater than ?i, then no form of the system C is of order greater 
than m + n — 2. 



CHAPTER VII. 

THE QUINTIC. 

115. To obtain the complete irreducible system of covariants 
of the quintic, we follow step by step Gordan's proof of the 
finiteness. Let us briefly recapitulate. 

The complete system of forms, which are not expressible in 
terms of covariants having a symbolical factor {ahy, is first 
found; this is called Aq, it is the complete system mod {aby. 
Generally A^ is used to denote the complete system mod (a&)^+^ 
To obtain the system J-^+i from the system A]c, a subsidiary 
system of forms Bjc is used. This system is a system of forms 
having ^ = (aft)^"*"^ a^^'^^'^^ hx^~^~* for ground-form. 

When the order of this form is less than n, Bk consists of its 
complete irreducible system. Otherwise if the order of (ft is 
greater than n we may take for B^ the single form <^ ; while when 
the order of <}> is equal to n, the system Bk consists of <f) and the 

n n n 

invariant (abf {bey {caf. 

Then it has been proved that the system A^+y^ may be obtained 
by taking transvectants of products and powers of forms from A^ 
with products and powers of forms from B^. 

116. The quintic will be written 

f=<^i = bx'= 

The system A^ contains /only. 
The system B^ contains only 

{abya^bi = H. 
The system A^ is then obtained from the transvectants 



115-116] THE QUINTIC 129 

If 7 > 2, this transvectant contains a term having a factor 
{hcf ; such a term can be expressed as a sum of symbolical 
products each containing a factor {aby, and is therefore = 0, 
mod {ahy. 

Hence we may reject these transvectants when 7 > 2, for all 
trans vectants which contain reducible terms may be rejected. 

The transvectant {H,fY contains the term 

{ahf {hcf aAcx' = U- (a^)' «x&a; (§ 51, Ex. (vi)). 

The system ^^ then consists of 

/ H, if, H) = t. 

The system By is built up from the form 

{aby a^ba: = i, 

this is of order < 5, hence we must take the complete irreducible 
system of the quadratic i. 

The system By^ then consists of 

i, {i, iy = A. 

The system A^ is now the complete system of forms for the 
quintic, it is made up of the transvectants 

U={f<^HHy, i^A'Y- 

Since A is an invariant we may suppose that e = (if at the 
same time we remember that A belongs to the complete system). 

Since H is a, form of even order, and i is a quadratic, all trans- 
vectants are reducible except those which have 

(i) /8 = 0, 

(ii) a = 0, 7 = 0. y3 = 1. 

Again t is the Jacobian of /and H, therefore 

^^=-M(/./)^ff^-2(/, Hyf.H+{H, Hy.p] 

s-^jy^'mod {ahy. (§ 78.) 

Hence any transvectant F, in which 7 > 1, can be expressed in 
terms of transvectants V in which the degree of the product on the 
left has been decreased and that on the right has been increased 
together with reducible terms (| 105, Cor.): for as we have just 
seen if /3 > 1 then U is reducible. 

G. & Y. 9 



130 THE ALGEBRA OF INVARIANTS [CH. VII 

Accordingly we have the following cases to consider: 

(i) a=l or 2, 13 = 0, y = 0, 

(ii) a = 0, /3 = 0, 7=1. 

(iii) a=l, yS = 0, 7=1, 

(iv) a = 0, /3=1, 7 = 0. 

All other transvectants are reducible or are expressible in terms 
of these. 

(i) a=l, 2, /3 = 0, 7 = 0. 

The irreducible transvectants are 

(/ 0, (/ *)^ (/ ^o^ (/ i'r, (/ ^J, (/^ ^T- 

To see that the other possibilities contain reducible terms we 
shall take one example. The transvectant (/^ i^y contains the 
term (/, i^ (/, i'Y. 

(ii) a = 0, ^ = 0, 7 = 1. 

Since t = - (aby (be) a^h^c^^ 

and (6c) (6i) c^i^ = \ {Q)cf ij" + {hif d - {cif K% 

the term — {ahy (be) (bi) a^ b^Cx^ix 

of (t, i) 

is reducible. Similarly the transvectants 

{t, ij, (t, i*y, (t, i*y 

contain reducible terms. 

Thus (t, i*y contains the term 

- (aby (be) (ai'y {ai") (bi") (ci"y (bi) c^H^ 
which is at once reduced by means of the above identity. 

We are left with the forms 

(t, iy, (t, iy, (t, ij, (t, i*y, (t, i^y. 

(iii) a=l, ^ = 0, 7 = 1. 

The only irreducible transvectant here, is 

(f.t,iy\ 

To see that the other transvectants are reducible it is sufficient 
to' remark that for example (/. t, i^)'* contains the term 

(/, ij.(t, iT 

/ 

/ ... 



116] THE QUINTIC 

(iv) a = 0, ^ = 1, 7 = 0. 
The trans vectants 

{H, i), {H, if, (H, i% (H, i% {H, t^/, (H, i^y 
prove to be all irreducible. 

We are left with 23 forms which are as follows* : 



131 



Degree 


Order 





1 2 3 


4 


5 


6 


7 


9 


1 










/ 








2 




i 








H 






3 


1 




ihff 




ih f) 






t 


4 


A 








ihff)' 




{i,H) 






5 




{i\f)' 




(iM? 








{i, ty 




6 






(S\Uf 




{z^Hf 










7 




{i\ff 








{i\ ty 








8 


{i\ Hf 




ii\Hf 














9 








{p, tr 












11 




{i\tf 
















12 


(,•5, /2)10 


















13 




ii', tf 
















18 


ii'Jtf' 



















* One very obvious remark is to be made regarding this, and all other complete 
systems obtained by the present methods. We are assured that every covariant 
can be expressed rationally and integrally in terms of those retained in the 
complete system, but there is nothing in the process to shew that the latter are 
all irreducible, except in so far as failure to reduce them may be taken as evidence 
in this direction. Theoretically then Gordan's process gives an upper limit to the 
irreducible system. 

The enumerative method, depending on the generating function, introduced 
by Cayley and finally developed by Sylvester and Franklin (Am. Jour. vol. vii.) 
gives a lower limit to the system and when the two methods give the same result 
the irreducible set has been obtained. The results even when identical have to 
be received with some caution on account of the enormous labour involved. 

9—2 



132 THE ALGEBRA OF INVARIANTS [CH. VII 

117. It is found that for discussing the properties of co- 
variants, it is convenient to have the indices of the transvectants 
which express them as low as possible. On this account it is 
usual to replace some of the forms in the irreducible system just 
given by others which differ from them by reducible terms only. 

In the first place the covariant 

is of fundamental importance in the quintic system. It is a cubic, 
and the system of forms for which it is a ground-form are 
irreducible when considered as forms belonging to the quintic. 

Now (H, i^y contains a term 

(aby(a{)^hiya^h = (j,jy, . 

accordingly we shall take the irreducible form of degree 6* and 
order 2 to be 

(i. jy = 'r- 
Similarly the form (t, i^Y may be replaced by {j, r), for (t, i^y 
contains a term 

(aby (be) (aiy (bi'y {cij a^ c^ 

= {{aby{aiy{bi'yaj>^, (c{"yc,') = (j, t). 
And (f^, i^y^ may be replaced by the invariant of j, 

(r, ry. 

This invariant will be denoted by C, the proof that it may 
be included in the system instead of (/^, i^y^ will be given later 
(§ 121). 

It will be found useful to denote the term 

(aiy (aiy (biy {bi'J (ai'O (6t'0 

of this transvectant by M. Then M may be taken as the invariant 
of degree 12. 

It may be recalled in fact in connection with the simultaneous system of a cubic 
and quartic (Gundelfinger, Math. Ann. Bd. iv.) that the two results originally agreed, 
but a revision of the generating function led to a reduction of the lower limit 
which it theoretically gives, and afterwards two forms included in the irreducible 
sj'stem as derived by the methods of Gordan and Clebsch were found to be reducible. 
The complete systems for the binary forms up to the octavic may be considered as 
accurately determined by the two methods combined. 



117-119] 



THE QUINTIC 



133 



Besides j and r, there is one more quadratic covariant, given 
in the list as (H, i^f. This is equal to {{H, iy, i); we may- 
substitute T for {H, t^y, and hence take as the remaining quadratic 
covariant 

{r,i) = -% 

118. The linear covariants. 

if, i^y = {aif {aHy a, = - ( j, if = a, 
(/, i'f = (ai)^ {aif {aif') 4" = (a, i) = - ^. 
{t, i*y contains the term 

{aby (be) {aiy {hij {ci'J {ci"J a^ 
= {{ahy{aiy(hi'ya^h^,a) 
= {r, a) = 7. 
(t, i'^y contains the term 
(aby (be) (aiy (bij {ai") {cif'J {ci^^'y i^' 

= ({aby (aiy (biJ (ai") ij'b^, a) - ^ (aby (aiy (biJ (ai") (bi") . a, 
of which the second term is reducible and the first 
= ((T,i),a) = -(^, a) = -S*. 

119. The invariants. 

(i,iy = A, 

{H,i'y = ((H,i%iy. 

The latter may be replaced by 
(r,iy=B. 
(f'\ i^y has been replaced already by M. 
(ft, i^y^ contains a term 

((/ i% (t, *■«)"). 
and hence may be replaced by (a, B) = — R. 

Taking the Jacobians of the 6 linear forms two and two, we 
obtain the 6 invariants 

(a^), (^7), (7a.). («s), (m> (75). 

* This is the definition of the linear covariant 5 of degree 13 given by Clebsch. 
In Gordan's book 5 = (r, ^)=-{S-, a) -^(i, t)'^ . a. In other respects the letters 
common to the two books are identical in meaning. 



134 THE ALGEBRA OF INVARIANTS [CH. VII 

The values of these invariants are* 

(a/3) = - (aiy {aij {ai") {WJ {h^^f {hi") = -M=- {{a)\ 
(/S7) = (id) (ra) (ir) = (^a)^ = (Ba) = R, 
{ya) = {ray = ]Sr, 
{aB) = -R. 

Now 8 = (({, t), a) = (it) (ra) 4 + i (^T)^ ct, , 

hence 

(^8) = {ia) (W) {i'r) (ra) + ^B . {iaf 

(78) = ((Ta)T^, {iT'){ia)r^-kB,a) 

= \{CM-BN). 

Also N = (7a) = {ray = {jif (j'iy (rj) (t/) 

= (>■) U'iyiv) {(ri) (jj) + (ri) (fi)}. 

Now from the theory of the cubic we know that any symbolical 
product which contains a factor (rjY is zero. 

Hence 

j^=(j.r)(ji)(Fnrj){Ti) 

= i(i?") (ri) {(ji) U'ij (Tj) - (fi) (jiy (t/)1 

= ^(i?"') (ri) {(jf) (*•*") (/O (rj) + (ji') (j'i) (fj) (ri')} 
= Hjjj (V) (rj) . (iij - i ((jjyj.j.', (ri) (ri') ij^'f 
= IAG- i(T. I [(tO^ . tV^ + (rt')^ . i.' - (Hy r,^]y 
= i^(AC-R). 

120. The third transvectant of / with j is identically zero. 
For 

(/ jf = - («x^ (biy Kj = - (aby {uy aj^ 

= + 1 (aby {(ai)' bj> - (biy a^] = \ (aby 4 . {(ai) b^ + (bi) a^} 
= (i,i) = 0. 

This property is sufficient to define j. For if yfr be an arbitrary 
cubic then (/, yfry is a quadratic. And in order that (/, yjry may 

* In future when no confusion can arise the comma between the two forms in a 
transvectant will be omitted. This is the usual practice, cf. Stroh. 



119-121] 



THE QUINTIC 



135 



vanish identically the coefficients of x-^, XjX^, x^ must be separately 
zero; giving three equations to determine the ratios of the four 
coefficients of i/r, — see Chap. xii. 

Again 

(/T)^=(«i)(«/)0}7«.* 

= - 2 {ajf (ajy a^j^jj, since {ajf a^ = 
= - 2 {ahj {acf (bif {cij aj)^c^. 

Now iahf {i'hy (cby =0 (§77). 

(aiy {hif A (ci) 

(acf {bey (i'cy 

(bay (i'ay (caf 

Hence, if S include all possible expressions obtained by inter- 
changing a, b, c, 

X(aby(biy(icy(cay 

= (aby (ciy (ci'y + (bey (aif (aij + (cay (uy (bi'y 

+ 2A.(ahy(bcy(cay. 
And therefore 

(/ r)'^ = - 2 (aby (acy (biy (ci'y a^b^c^ 

= - (aby aj}^ . (aiy (ci'y c^-^A. (aby (bdy (cay a^b^c^. 
But (aby(bcy(cayaa,b^C:, 

= (aby (bey (ca) [ — (be) a^ — (ab) c^ a^Cx 
= - (ahy (bey (ea) a^c^ - (aby (bey (ca) a^e^ 
= \ (bey (ea) (ab) aj^ + \ (aby (be) (ea) ei 

= -(ify=j- 

Therefore 

(f,Ty = -i.a-lA.j. 

121. To obtain the relation between the invariants C and M, 
we take the expression for M and introduce in it so far as possible 
symbols referring to the cubic j and its Hessian t, for 

G=(Try. 



136 THE ALGEBRA OF INVARIANTS [CH. VII 

Now 
M = (iaf 

= {3ir{j'iJ{F){j'i") 

= (» im [{ja) (ih) - ijb) {ia)Y [(/a) (i'h) - (j'b) (i'a)f 
= (» (/&) [ - 2 (ja) (jb) (ia) iib) + ijby {ian [{j'ay (i'bf 

-2(fa)(fb){i'a)(i'b)l 
since ( jaf ai = 0, and (/ft)* b^ = 0. 

In this expression we will introduce, as far as possible, symbols 
referring to ; ; for 

{iaya^^=j = {i'byb^\ 

M is then seen to be the sum of four terms, viz. 

W{j'h){jbnianj'af(i'bf 

= ijj") UT) UfJ UJ'J = - (tt)'' = - C : 

4 {jay ij'by (jb) (fa) (ia) (i'b) (i'a) (ib) 

= 2 (jay (j'by (jb) (j'a) [(iay (i'by + (iby (i'ay - (uy (aby} 
= ^(jj'J(jj"J(jj"'){j'j") 

- 2A . (jay (j'by (aby [(ja) (j'b) - (jj) (ab)] 

= - 4C + ^ . (jjj (aby {(ja) (fb) + (jb) (j'a)], 

since (jay ai = 0, 

= - 40+ 2A . (rt7= - 4(7 + 2AB : 

- 2 (jay (jay (jb) (j'b) (i'by (ia) (ib) 

= 2 (jay (fay (jf) (ff) (ia) (if) 

= (i«) (fa) (ia) dj") [(jay (ffj + (fay (jj'J - (fay (jfy] 

= - (ray (fay (ia) (if) = - (ra) (rf) (fay (iay = C : 

the last term of M 

- 2 (jby (j'by (ja) (fa) (iay (i'a) (i'b) 

is obtained from the one just considered by interchanging i and i', 
a and b, its value is therefore C. 

Hence M = -G - 4>G + 2AB + 2G 

= -SG+2AB. 



121-122] THE QUINTIC 137 

122. The covariants of orders 0, 1, 2, with the exception of i, 
have been replaced by transvectants, of index not greater than 
2, of simpler forms. We may simplify the expressions of the 
others in the same way, in fact this has been done already for 
two of the covariants degree 3, viz. j and (j, r). The remaining 
cubic covariant (/, i^f = ((/" iy, i) 

The transvectant (H, if may be replaced by its terra 

it will be convenient to write this covariant 

The remaining quartic covariant is 

and may then be replaced by (p, i). 
Now since (ajf a^ = 0, 

{aJT o,xjy = (oy? CLxjxay 
= i {{ajy ai^y + 3 {a^y a^ix(^y\ 

Hence 

{p, i) = {ajf (ji) a^% = {ajf (ai) aij^i^. 

Now ' (/,«)=(/-(>?>) 

= - (aj) {(«*)> - (aj) 4}' O'i 
= 2(aj)2(ai)4ja,a«^ 
= 2 {p, i), 
which gives another expression for the same covariant. 

For order 5 we have only to consider the transvectant {t, i^y ; 
this has a term 

(ahy (be) (dy {aij a^b^'cj", 

hence this transvectant may be replaced by {{H, iy, {/, iy), and 
therefore by 

(P'j)- 
Lastly the covariant order 7 may be replaced by the Jacobian 



138 



THE ALGEBRA OF INVARIANTS 



[CH. VII 



123. To express any transvectant of two covariants of the 
quintic in terms of members of the irreducible system, it is in 
general advisable to use as far as possible symbolical letters 
referring to covariants such as j, i, t, ^ etc. instead of those 
belonging to the quintic itself. We give here the values of some 
of the trans vectants, partly for the sake of reference, and partly as 
examples; we would recommend the student to verify a few of 
them*. 

The following table gives the 2nd transvectants and Jacobians 
of the quadratic and cubic covariants. 



2nd 
Trans- 
vectants 


Jacobians 


i 


r ^ 


J U, 


U, r) 


i 


A 


5 


^{Bi-Ar) 


-U,i) 


lai-¥\Aj 


^ar + \Bj 


T 


B 


G 


hiCi-Br) 


-U,r) 


\ar + \Bj 


hCj 


s 








J}^ 


— \aT 


l^^+hi 


hr 


J 


— a 





-y 


T 


-h^T-ya 


-h" 


u, *■) 


i/3 


-§y 


i«-i^a 


-i^ 


-J{2a2-fylr-3i5i} 


-i{3yJ + a(J,r)} 


1 


y 





-iCa 





IBr + iCi 


\Ct 



The third transvectants of the cubic covariants are 

(j, ( j, i)f = B, (j, (j, T)y = G, {{j, i), (j, r)y = 0. 
In obtaining the values of Jacobians it is well to remember 



the formula 



m — n 



proved, Chap. iv. § 77 ; where /, <f> are binary forms of orders m, n 
respectively ; and the order of each of the forms /, ^, i/r is not 
less than 2. This will be of frequent assistance, since of the 15 



• Should he experience any great difficulty he will find some of them worked 
out in Gordan's Invariantentheorie and Clebsch's Bindren Fortnen. 



123] 



THE QUINTIC 



139 



«o 


1 
+ 


1 
i 

1 

1 


1 

s 


1 


r 


1 

+ 

+ 

Hw 

1 


1 

s 


Hn 
+ 

oq 

r-te 
1 

H« 
+ 


1- 

f ■* 

+ t 

r^« HN 
1 1 

05 


?^ 


+ 


1 


1 


1 

H<N 


5 

1 

"6 

Hn 


+ 


8 


He 
+ 

cs 

H<N 


o; 

H» 

1 

He 
+ 


«1 


Kin 
1 

> 

+ 

1 


B 
1 


Him 

1 

I 


1 

HN 


1 

HN 
J 


HN 

+ 

1 


1 

HN 


+ 

B 
'^ 

HN 

1 
•<^ 

H« 

1 

05 
HN 


1 

He 

+ 

Hm 

1 

HM 


e 




<«. 


?«• 


«o 


1 


1 
HN 

1 


1 

HN 


HM 
1 

+ 

1 


HM 

1 

+ 

8 

HN 

1 

B 


"^ 


•- 


(- 


Sf} -^ 




a. 


•5 



140 THE ALGEBRA OF INVARIANTS [CH. VII 

irreducible covariants of the quintic for which the order is not 
less than 2, 9 have been expressed as Jacobians. 

It is useful to know the values of the following transvectants 

124. For purposes to be presently explained the transvectants 
(/. cf^ (/. '^^y, ••• will be required. Such transvectants are 
calculated step by step, first (/, a), then (/, a*)'' = ((/, a), a), 
and so on. 

• It will be useful then to know the values of the transvectants 
of certain of the covariants with a, f3, y, 8; when these are known 
the values of such transvectants as (/, a')' may be obtained with 
great ease. 

125. Syzygies. It has been proved (§ 77) that the product of 
two Jacobians, or the square of a Jacobian can be expressed as a 
sum of terms, each term being the product of at least three forms. 
Now nine of the covariants of the quintic are Jacobians, — we 
must exclude the forms (la), (ra), (^a), (a/3), (aS) for one at least 
of the quantics in each of these Jacobians is of order less than 
2 : hence we have 45 syzygies. 

A more general method of obtaining syzygies is given by 
Stroh. He considers four different forms, and seeks to obtain 
the syzygies which are of unit degree in each of these forms. 
First for three forms there is evidently only one such relation, 
that for weight unity 

/i (AA) +Mfs,A) +/s(A,A) = 0, 

this is written for short { f 1/2/3) = 0. 

The other syzygies obtained by Stroh arise from the symbolical 
relation 

(ab) Cxdx 4- {cd) axhg. = {ad) hxCx + (c6) axdx , 

which may easily be verified. Raise both sides of this identity to 
the power i and expand by the binomial theorem : hence 



S iV) {abf(cdy-^aJ-^bx^-^Cx'dx' 
= 2 (l) {ady {cby-^aj-^dj-^cx'^bx- 



123-125] THE QUINTIC 141 

Hence 

This is written 

Other syzygies may be obtained from it by interchange of the 
various quantics concerned. But it will be noticed that (fif 2/3/4)% 
is unaltered if one pair of letters is interchanged and at the 
same time the other pair. Also this expression is only changed 
in sign i( / and / are interchanged. Hence only three distinct 
syzygies of weight i are obtained in this manner. 

If z = 2 

(////). ^ (//f//+(//r// - {//)%/ - (//y// 

Whence 
{////}. ^ i [(////% + (////). + {////%] 

= (//) (//) + (//) (AA) + (/a/4) (//s) = 0, 

a result already well known. 

Also 
[/A//1 = i [(////% - (//A/% - (////a).] • 

= (//T// + (/3/yy./ - (//)%/ 

- i//y// + 2 (//a) (//) = 0. 
This is the same relation as that given (§ 78) for the product 
of two Jacobians. It will be seen at once that the other syzygy 
of weight 2 is deducible from this. 

The syzygies of higher weight may often be simplified, in the 
same way. 

The syzygy 

{///a/.}.= 

is remarkable from the fact that the forms in it need not be of 
order higher than unity ; while in the syzygy 

from which it was deduced, each of the forms is of necessity of 
order 2 at least. 



142 THE ALGEBRA OF INVARIANTS [CH. VII 

In general in the syzygy 

(/iA/s/.)i=o 

each of the forms/1,/2,/3,/4 must be of order i at least ; but from 
these syzygies others may be deduced which are true for forms of 
lower order. Stroh, in his papers on syzygies*, deduces many 
such. 

We will obtain such a syzygy from that of weight 3, 

(/l/./3/4)3=0. 

If \, fi, V be three quantities whose sum is zero, we know that 

hence 

3 {ah) (be) (ca) aj)^c^ = {hcf a^" + {caf b^^ + {abf c^\ 

We thus obtain 

h [(a^'h'c^'d^% - {(ix'cMz'b^% + {a^'d^'b^'c^^),] 

- (ab) (be) (ca) aJ)xCxd^ 

+ (b.\ cj) (a^\ d^J + (c^^ a^') (b^\ d^J + (a^\ b^') (c/, d^'Y = 0. 

It will be seen at once that a^, b^, Cx, dx are factors of this 
syzygy, we may then divide by any one or all of these factors ; or 
else we may multiply the syzygy by a power of any one of them. 
The syzygy is then true whenever the order of each form is 
gi-eater than unity. It is not difficult to see that this derived 
syzygy is just as general as the original one 

in other words, the original syzygy might be derived from it. 
Stroh writes the syzygy just obtained in the form 

{fHq% = ( fqf iH) - (fqJ i<f>q) - (f<f>y (qq) +A(qq') <!>)' = o, 

where q, q' are quadratics, and /, (f) are forms of order 2 at least. 
Again from the syzygy 

* Math. Ann. Bd. 33, pp. 61-107 (§§ 18-22) ; Bd. 84, pp. 306-320, 354-370 ; 
Bd. 36, pp. 262-288 ; in § 3 of this latter paper he gives a list of syzygies of the 
kind just mentioned which he has deduced. 



125-126] THE QUINTIC 143 

we obtain 

{ffH\ 
=p i<}>cf>y + 6 (fff i<fy<f>y + ^^ (f/y 

- 2f<f> (fcf>y + 8 (f<f>y (/</,) - 6 [{f<i>yY = o. 

In this write i- for (f), where i is a quadratic. 
Now 

(i\i:^y=i[(nyy, 
(i\i^y=i(ny.i% 
2(f,ijif,i^) = 2i{(f,iy,i).(f,{) 

—see § 78. 

Hence substituting these values in the syzygy obtained by 
writing i^ for <f), and then dividing the result by 2i^, we obtain 

=f{py - % {{fiy/y - [{fiyj + (n)-^ {f/y + ^"^ {ffy = o. 

From this may be obtained a syzygy 

by means of the operator (-v/r >j. This operator requires 

that i/r and / should be of the same order, but when the syzygy 
is written symbolically it will be seen at once that factors of the 
form ax and h^ may be introduced so that the relation is true 
whatever be the orders of / and i/r, provided that neither is less 
than 4. 

In the same way we may obtain a syzygy 

where both i and r are quadratics. 

126. Application to the quintic. The forms of the quintic 
consist of /, H, i, j, r, p, a, /3, 7, B and some of the Jacobians of 
these forms. 

A large number of syzygies may be at once obtained by 
writing for the forms in the general syzygies of the last paragi-aph 
covariants of y. 



144 THE ALGEBRA OF INVARIANTS [CH. VII 

We shall content ourselves with a few examples. 
{fiH)=f.{iH) + H{fi)-it = 0. 
(fjH)=f.(jH)-H(jf)-jt = 0. 

[fM^ = i(M' + Hi-^ + 2A>- +PA = 0. 

[ijri]^ = B (ji) - A (jt) - a^ + iy = 0. 
(ppii)i = -Bp + fm^ - ^Air - ^Bi^ -t'+^AJ(z= 0. 
{a^ryB}^ = \M{BN - CM) - \N{AN- BM) -B? = Q. 

From this last we deduce the relation connecting the irreducible 
invariants A, B, G, R. 

It is eas}^ to write down a great number of syzygies in this 
way. Stroh {Math. Ann. Bd. 34, pp. 354 — 370) has given a list 
of 168 syzygies of the quintic. The notation for the elementary 
syzygies given here is not quite the same as that used in Stroh's 
paper, but the notation is there explained ; the notation here has 
been mainly taken from a later paper by the same author (Stroh, 
Math. Ann. Bd. 36, pp. 262—288). 

127. Reducibility of syzygies. If 

be any syzygies, and if Pj, Pg. ••• he any products of forms such 
that Pi^S^i, Po'S'a, •.. are expressions all of the same degree and 
order, then 

is a syzygy. In this way it will be seen that an infinite number 
of syzygies may be built up. 

^ syzygy 8 = 

is said to be reducible, when 

/S=P,5fx + PA+-., 
where 8^ = 0, 8^=0, ... are syzygies whose degree is less than 
the degree of *S = and the P's are covariants. Otherwise it is 
said to be irreducible. 

It will be seen at once that any syzygy which contains a 
product of only two irreducible forms must be irreducible. All 
the irreducible syzygies which have yet been found for the quintic 
are of this nature. 



126-129] THE QUINTIC 145 

128. It may happen that certain syzygies 
^, = 0, ^^2 = 0, ... Si = 

are such that certain products of forms Pj, P^, ... Pi may be 

found, for which 

P,8, + P,S, + ...+PiSi=0, 

the expressions P, S being regarded as functions of the con- 
comitants — which for the moment are treated as independent 
variables. 

Such a relation is called a syzygy of the second kind. 

The following is an example, 

(/1/./3/4) =/i (fJsA) -A (/1/3/4) +/3 (/1/./4) -A (AM) = 0. 

Thus for the quintic 

(fHvj) ^/(Hij) - H ifij) + i i/Hj) -j if Hi) ^ 

is a syzygy of the second kind. 

Syzygies of the second kind may clearly be reducible or 
irreducible. Between them may arise syzygies of the third kind, 
and so on. 

The following questions at once present themselves. ' Is the 
number of syzygies finite when the system of forms is finite?' 
' When the syzygies of the first kind are finite in number, are also 
those of the second and of higher kinds finite V 'Is there any 
limit to the number of kinds of syzygies which arise from a finite 
system of forms ? ' 

All these questions have been answered in the affirmative by 
Hilbert {Math. Ann. Bd. 36, pp. 473—534). They are partly 
considered in Chapter IX. 

129. The typical representation of the binary quintic. 

For special purposes, some particular linear transformation of a 
binary quantic may have peculiar advantages. Thus any particular 
term of the quantic may by a special transformation be made to 
vanish. If the quantic has two linear covariants, such that the 
determinant formed by their coefficients does not vanish, these 
may be taken for the variables : the transformed quantic will then 
possess the property that every one of its coefficients is an in- 
variant. We proceed to prove this. Let a^, ^^ be two linear 

G. & Y. 10 



146 THE ALGEBRA OF INVARIANTS [CH. VII 

covariants of the quantic a^^, which are such that (ay8) is not 
zero. Then raising the identity 

(a^) a^ = (a/3) a^, - (aa) ^^ 

to the nth power, we obtain 

(a/3)« . a^« = (a^y . a^« - n {a/3y-' (aa) a««-^/8^ + . . . 

The expression on the right is the transformed quantic, and 
from the symbolical form of the coefficients, it follows that they 
are all invariants. 

For the general quintic any pair of linear covariants may be 
chosen ; for example those which we have written a and /3. 

The coefficients may be easily calculated with the help of the 
table given on p. 139 ; they are as follows : 

(/, a'^y = - UN- ^AM + \~^ 31 +\^ {2BM-AN) = (m. 
\A' 



{f,a^y = -AR(M+~^ 



{f,^y = -M' + ^AM{AN'-BM)-\-fi^. 

Further the invariant (a^) = — M = SC— 2AB must not be 
zero. 

To be more accurate the coefficients given above should be 
divided by (— My. In the expression for any covariant in terms 
of the actual coefficients, the above transformed coefficients may 
be substituted, the covariant multiplied by a power of the 
determinant of transformation is then equal to the expression 
thus obtained. In this way any covariant may be expressed in 
terms of the invariants and two of the linear covariants. 

To illustrate a different method of expressing any covariant 
in terms of the invariants and two linear covariants we shall 



129-130] THE QUINTIC 147 

obtain j in terms of the invariants and the covariants a. and 3. 
Raising the identity 

Wjx = (j8) ax - (» ^x 
to the third power 

_ R^j = (j, 8-^y a'-S (j, B^af a^S + 3 (j, Sa^f aB^ - {j, a'f S^. 
But by the method of § 124 

{j,aB^) = -^NE, {j,hJ = -^R{GM-BN). 

Therefore 

B?j = ^{GM- BN) Q? - %Nci^B - B\ 

130. Given two binary forms of the same order, in particular 
two quintics, can one be linearly transformed into the other, and 
if so how ? 

The reply (in part) to the first question is that if the absolute 
invariants of the two quantics are equal to one another, and if 
they each possess a corresponding pair of linear covariants of 
which the determinants do not vanish, then the quantics are 
transformable into each other. The question will be found dis- 
cussed in Clebsch, Bindren Formen, § 92 ; and for the case where 
there are no linear covariants in § 105. 

When two quintics have equal absolute invariants and one 
of the 6 invariants (a/S), (ay) ... is other than zero, say (a/3), we 
may transform one quintic into the other thus : — 

Let unaccented letters refer to one quintic, and accented letters 
to the other ; we transform each quintic, so that the variables in 
the first are a, /3, in the second are a, yS'. 

Thus f=AoaP + 5A,a*^ + 

f = Ao'<x'' + 5A,'a:'B' + 

A 
Let the ratio -r? = r, then since the absolute invariants for 
A 

the two quintics are equal it follows that 

A^~ A'^' 

and hence | = r=. 

10—2 



148 


THE ALGEBRA OF INVARIANTS 


Similarly 


^, = r3, and -^, = r^ 


Hence also 


<■ 




A ' ' ) A ' ' > A ' '» 

.n.Q Jlj M.2 




:ttl-r~^ ^*-r-« ^-r~^ 
jO-s Ji-i Ji-s 



[CH. VII 



The quintic /' may now be transformed into / by means of 
the transformation 

6 

a = T^a', 

/3 = r^^^'. 



131, Associated forms. If yi, y^ is a pair of variables 
cogredient with aj^.a;.^; then the two forms 

are invariantive. Now regard x for the moment as a constant, 
and the two equations just written down as equations of linear 
transformation to transform from the variables y^, y^ to new 
variables f, 77. The variables of the transformed form are co- 
variants, hence its coefficients are invariants — or to be more 
accurate co variants, for they contain a; but not y. Let us proceed 
exactly as in § 6. 

The determinant of transformation is 

(I'Z) = «."=/• 
From the identity 

we obtain the transformed quantic 

i^Y 6/ = {h7)Y |« - n (bvT-' (6f ) f«-^ 77 + (I). 

Let us calculate the coefficients of the transformed form for 
the case of the quintic. 

(bvY m' = hj^ iha) a,* {ha') aV = ^H .f, 
(brjy {b^y = b,^ (ba) {ba') iha") a,*a',*a\* 

= ^b^^ (ba) a^*a'^'a"a? {{haj a'V + (ba'y aV - {a'a'J b^'] 

= -t.f-^H. {ba) b,*a^* = -t.f, 



130-131] THE QUINTIC 149 

ihrj) {b^y = b^ (ba) (ba') {ba") {ba'") a^^a'^'a^^a'^* 
= {baj {ba) {ba") h^aja^*a"'^' . aV 

-^H .{ba){ba'")b^'a^'a'"^* 
= {bay {bay b,a7 aj J' - h H'f 

{b^y = (6a) {ba') {ba") {ba'") (6a»^) ax*aVaVa"Va'V 

= {ba'y {bay {ba}^) a'^'aj'a}\* .p + Etf. 
Now {{bayi^o:yb^aia'i,f) 

= j {bay {ba'y {{be) a^ali + 3 {ac) b^aiali + 3 {a'c) b^a^a'^^] c^* 
= {bay {ba'y {be) a^'a'a^c^" 

+ ^{aby{ba'yaJ'a'^Kf 
= {bay {ba'y {be) ai a'^'c^* + f {aby {{a'b) a^ + {a'a) b^\ aV • / 
= {bay {bay {be) a^'aHe^' + f (/, i) ./. 
But the transvectant 

{{bay {bay b,aio!.\f) = {\ifj) = ^ (/ i) ./ 

Hence (6a)- {bay {ba}'') aia'^^e^'' 

= -{f,i)-f> 
and therefore 

{b^y = -{f,i).f + Htf. 

The transformation is then 

+ o.[ir-lH^)^V' + {{f,i)f^-Ht)r,^ 

Now let O {y) be any covariant oi f{y), then when the above 
transformation is made, the coefficients of f{y) are replaced by 
the corresponding coefficients of the powers and products of ^, 7/ 
in the expression on the left. Let O thus transformed become 
^' (^> "nX then ^ is equal to ^' divided by a power of / the 
determinant of transformation ; thus 



150 THE ALGEBRA OF INVARIANTS [CH. YH 

This equation is an identity. We may replace in it y by ar ; 
when this is done, ^ becomes /and 17 becomes zero, hence 

- . , <!>'(/ 0) 
^(^)= ^x ' 

Hence any covariant of the quintic is equal to its leading 
coefficient, when the original coefficients of the quintic are replaced 
by the corresponding coefficients in the form (I), divided by some 
power of f. 

From this we see that all covariants of the quintic may be 
expressed rationally in terms of the covariants fy H, x, t, (/, t), 
in such a way that / alone occurs in the denominator. Such a 
system of covariants in terms of which all covariants of a system 
may be algebraically expressed is called a system of associated 
forms. We have confined ourselves to the case of the quintic, the 
results obtained are however true in general. The coefficients of 
the transformed quantic may always be expressed as rational 
integral functions of/, the covariants of degree 2, and the Jacobians 
of these latter with / And this is in fact the simplest system of 
associated forms. 

The matter will be found fully discussed in Clebsch, Bindren 
Formen, ch. vii. The student who requires further information 
on the subject of typical representation will find it in the chapter 
just quoted and the two succeeding chapters of Clebsch's book. 

The reduction of the quintic to a sum of three fifth powers will 
be discussed in Ch. XI., and so nothing need be said on the subject 
here, especially since it concerns the non-symbolical treatment of 
the subject rather than the symbolical treatment. The special 
canonical forms to which the quintic may be reduced, when one 
or other of its invariants vanishes, will be found in Prof. Elliott's 
Algebra of Quantxcs. 

For a symbolical treatment of the subject the student is 
referred to Gordan's Invariantentheorie, or Clebsch, Bindren Formen, 
§§ 93—96. 

132. The Sextic. The difficulty in obtaining the complete 
irreducible system of concomitants of a binary form increases very 
much with the order. The system for the sextic is obtained here ; 



131-132] THE QUINTIC 151 

it affords examples of a method of reduction applicable to forms 
of a higher order, but not required when dealing with the quintic. 

The arrangement in systems of forms whose grade does not 
exceed a certain number is followed as before. 

The system A^ contains only /; Bq contains only 

The system Aj consists of 

/, H, (/ H) = t. 
For Bi we must take the complete system of 

Now i is a quartic, and its complete system is 

To find the system A^ we must take the transvectants of 
powers and products of forms of A^ with powers and products of 
forms of i?i. 

Now the form (i/Y can be shewn to vanish, for 

(ify = (abY(bcy{ac)a^c,' 

= — l{ah) (be) (ca) [{aby (be) a^ci + (bcY (ca) b^aj^ 

■^{cay{ab)cJ)J''\ 

= l{ab) (be) (ca) [(aby cj- + (bcY a^' + (caY 6^*], 

on using Stroh's series 

/ / / 



( / J / \ 
V 1 1 1 A 



But (aby (be) (ca) c^* = 0, 

since it changes sign when a and b are interchanged. 

Hence (i/Y = 0. 

The quadratic covariant (i/Y is of great importance ; it is 
usually denoted by the symbol I. 

If any covariant can be expressed as a symbolical product in 
which the factor (iay appears, it can be expressed as a sum of 
transvectants of I with other forms. For such a covariant 

= ((iay i^aj, 0)p + 2 ((iay aj, <!>')'> 

= s (I, <^y. 



152 THE ALGEBRA OF INVARIANTS [CH. VII 

Again 

A = {i, if = {iaf (aby iA' + X (aby . i. 
And 

{{iay i^a^\ b^J = i {iaf bj> {{abf 4 + 3 {aby (ib) a^} 
= (iay (aby i^b^^ + 1 (iaf (aby b^\ 
but (iay i^ai = 0, 

hence (iay (aby i^b^^ = - 1 (iay (aby 6a,' = - f (l/Y- 

Now since a and b are equivalent symbols 
(iay (aby i^bx' = - ^ (aby ij" {(iay ba>^ + (ia) (ib) a^b^ + (iby aj] 
= - f (iay (aby i^^' + I (aby . i, 
and therefore ^ = i i{fy mod. (abf. 

Thus every form of Bi except i and the invariant (i, iy 
= modd. (iay, (aby. 

133. We shall first find the system which is relatively 
complete with respect to the moduli (aiy and (aby (§ 107, Cor. II.). 
This is obtained by taking the transvectants of powers of i, with 
powers and products of forms of the system A^. Let us call this 
the system G. 

First consider the forms 

(i\P)y. 

We have (i,f), (i,fy- Every other one of these forms 

= mod. (iay. 
Next consider forms 

(i\ m)y. 

We retain only (i, H), for (i, Hy contains the term 

(iay(abyi^^a^'b^'; 

this can (§ 63) be linearly expressed in terms of covariants 

(iaya^^b^\ (ibyb^^a^\ (iay (ab) i^ax^b^\ 

(iby (ab) ij)^a^, (aby a^b;^ . 4*. 

Hence (t, Ey^Xi''^- id ./. 

The form {i, Hy contains the term 

(iaf (aby i^a^b^* ; 



132-134] THE QUINTIC 153 

and the form (i, H)* contains the term 

hence these may both be rejected. 

All other forms (i*, H^y contain a term having a factor {iHy. 

No one of the forms (i", ^)t need be retained, for if 7 = 1, the 
transvectant is the Jacobian of a Jacobian and another form and 
therefore reducible (§77). 

If 7 > 2, the transvectant always contains a term which involves 
a factor {ia)^ or a factor {idf, and therefore which 

= mod. {aif. 

If 7 = 2, the transvectant {i, ty contains the term 

= {{i,H)\f) mod. {iHY 

= \ (^^ /) mod. (aiy 

= \i {i, f) mod. {aiy. 

The general transvectant 

{i\ f^Hyfy 

may be treated in the same way. If e > 2, the transvectant 
contains a term which 

= mod. (aiy. 

And if 6 :}> 2, it is certainly reducible, except for the cases 
already discussed. 

The system C then contains the forms 

/ H, t, i, a, ^•)^ (/ i), {H, i), if, iy. 

134. To find the system A^ we must now take all possible 
transvectants of powers and products of the system C, with powers 
and products of the complete system of I. 

Now I is a quadratic, its complete system must then consist of 

I, {I, ly. 

The invariant {I, Vy is the same as an invariant already found, 
viz. : 

(t,A)^ = Hi Wy mod. (a6)« 
= \{liy mod. (a6)«. 



154 THE ALGEBRA OF INVARIANTS [CH. VII 

Since Z is a quadratic and all the co variants of C are of even 
degree, we need only consider the transvectants of powers of I 
with each form separately. 

The forms 

a/), {i,fr=^\ {i\f)\{iM)\(i\fr,{i\fr 

are all irreducible ; so also is 

{I, H). 
The covariant (l, Hf contains the term 

= 2{(i,iy,fy 

= \ (iij {i'af i^'a,' + fi (i iyf. 
But the term 

is linearly expressible in terms of the covariants (§ 63) 

(nj ««,«, {iiJ (i'a) i^a^\ (ia)* a^ i^\ 

{iay{ii')i^^a^, {i'a)* a^H^*, 

each of which is reducible. 

Hence {I, Hf = \^ (nyf+ \li. 

Now, if )8 > 2 

(^, Hy 

= ((I, Hy, i-^y-^ = (uy (D + x^ {H, ^-^/-^ 

hence these transvectants are all reducible. 
The covariant (I, t) is reducible, § 77. 
The covariant {I, ty contains the term 

i{i,sy,fy 

and is therefore reducible. 
The covariant, yS > 2, 

(/a, ty = {{I, ty, t-'f-^ 

and is reducible. 

Hence all the covariants 

{i\ ty 

are reducible. 



134] THE QUINTIC 156 

The covariants {I, i), {I, if, (^^ if 

are irreducible ; but 

{l\i)* = {l\{ahfa,'h^^f 

= ({l,fy, (IJffmod.iabf 

= 4 ((t, if, a iff mod. {ahf, 

which is reducible when considered as an invariant of the 
quartic i 

The form (l, (f, i)) is reducible, § 77 ; 

(I, {f, i)f contains the term 

• {{l,ff,i) = {^,i) = -v; 

(^^ {f, i)f contains the reducible term 

{l,{{l,f)M))- 

Similarly {l\{f,i)f, (l\(f,i)y 

may be reduced. 
The forms 

(z^(/,^•))^ (/^ (/,*■))«, (^^ (/,*•))« 

are however irreducible. 

The form (I, (/, if) is not reducible. 
Now (I, {/, iff contains the term 

(ii,ff,iy=2{A, if = ii(iif... 

(see § 89). 

Hence (Z", (/, iy^y is reducible when /3 > 2, for this 

= ^ (iif it-\ if-^ + X (Z«-i, l'y-\ ' 
Lastly (l, {H, i)) is reducible by § 77. 
{I, {H, i)f contains a term 

((I, Hf, i) = X, (iif (/. i) + X, {li, i) 
and • {l'^,(H,i)y, /3 > 2, 

= (i-\(i,{H,i)fy 

which IS reducible. 



156 



THE ALGEBRA OF INVARIANTS 



[CH. VII 

Thus the system A.^ contains the forms of the system G 
together with /, (I, I)- and 

(i,n (ijy> {i^/y, (^^/)^ (^^/)^ (^^/)^ 
(I, H), (I, i), {I, if, {i\ If, (I, (/ ^))^ (/^ (/ i)y, 

{i\{f,i)r,{i\{f,i)r,{i,{f,in 

The system B^ contains only the invariant (/,fY', we merely 
add this to the system A2, and the result is the complete system 
for/ 

We append the following table giving the 26 irreducible 
concomitants of the sextic. 



Degree 


Order 











2 


4 


6 


8 


10 


12 


1 








/ 








2 


if,/)' 




iff)*=i 




if,ff=s 






3 




(/, ir=i 




(/, ^7 


if^l 




if, S) = t 


4 


{i, ly 




if If 


if I) 




{H,t) 




5 




ih I)' 


ii, I) 




(II, I) 






6 


{I, If 






((/. if, I) 
((/. i). If 








7 




if, i^Y 


if I'f 










8 




ii, I'r 












9 






iif,i),i')' 










10 


if, I'f 


if ^)* 












12 




iif,i),l')' 


• 










15 


Of, i), i¥ 















Ex. (i). Prove that if the covariant a, of a quintic /, is identically equal 
to zero, then also 

/3=0, 7 = 0, 8 = 0, 3 = 0, ipi) = 0, M=0, M=0, R=0, 

Bi = AT, {B-%A^)J=0, Bf+yr=0. 

Ex. (ii). In the Iqst example either 

j=0 or B = IA^. 



184] THE QUINTIC 157 

Prove that in the former case ever}- covariant vanishes with the excep- 
tion of 

and that these are connected by the relations 

At + \i\fi) = 0, 
2[(/,^•)P + ^^•2+/2J=0. 

Ex. (iii). Prove that if o vanishes identically and^' is not zero, then 

B^IA\ t^lAi, A{Af^^ij)^0. 
The latter result gives an alternative, but if 

then Aj=^ (ij, if=%Aj. 

Hence in either case since ^' is other than zero, 

^=0, B=0, C=0. 

Shew further that j is a perfect cube : that it is a factor of / : that i is a 
factor of j : that p is a, perfect fourth power, having J for a factor : and that 
jt? is a factor of H. 

Ex. (iv). If all the invariants of a quintic vanish shew that a must vanish 
and that j must be a perfect cube and a factor of /. 

Ex. (v). If r, = {xi/), i = a,^-^a„ f=a^^, 

then 

(i) n = % 

f-fi2/) = e + WJf-r,. 
(ii) n=Z, 

(iii) «=4, 
(iv) 71 = 6, 

{aehsch). 

[The student, who wishes for information concerning special quintics — 
when some of the invariants vanish — is referred to Clebsch, Bindren Formen, 
ch. VIII., and to Elliott, Algebra of Qualities, ch. xili.] 



CHAPTER VIII. 

SIMULTANEOUS SYSTEMS. 

135. It was proved in Chap. vi. § 103, that if Si, 82 be any 
two finite and complete systems of forms, then the system S 
formed by taking transvectants of powers and products of powers 
of forms of Si with powers and products of powers of forms of ^S^a 
is both finite and complete. If S-^, S2 be the complete systems for 
any two binary forms fx^fi', then S is the complete system of 
concomitants for the forms yi, ^a taken simultaneously; for S is 
complete and contains both f^ and f^. Hence the complete 
irreducible system of concomitants of a pair of binary forms is 
finite. 

136. To make the matter clearer, let us briefly recapitulate 
the argument. 

(i) Any concomitant of the simultaneous system can be 
expressed as a sum of symbolical products ; the factors in which 
are all of the following types 

(a6), (a^), (aa), a^, a^: 

where letters of the Roman alphabet refer to the quantic /j, and 
letters of the Greek alphabet to the quantic^. 

(ii) Any concomitant of the simultaneous system can be 
expressed linearly in terms of transvectants of products of forms 
belonging to the complete system for/j, with products of forms 
belonging to the complete system for f^. 

For any symbolical product in which the letters are partly 
Roman and partly Greek is a term of a transvectant ( U, Vy, where 
U is &, product containing only Roman letters and V a product 



135-137] SIMULTANEOUS SYSTEMS 159 

containing only Greek letters. But by § 51 any term of the 
transvectant {U, Vy is equal to 

{U, vy-¥t\{U, vy, 

where U, V are obtained by convolution from U, V respectively ; 
X. is numerical and p is less than p. 

Now U, U axe covariants of/j and hence maybe expressed as a 
sum of products of the irreducible forms of /i ; similarly V, Fmay 
be expressed as a sum of products of the irreducible forms of^. 

Hence the theorem is true for any symbolical product, the 
letters of which refer some to/i and some to/g : and therefore it is 
true for any concomitant of the simultaneous system. 

(iii) The system of transvectants {U, Vy, where ?7 is a 
product of concomitants of/i and V a product of the concomitants 
of /a, is both finite and complete. This was proved in § 103. 

137. The complete irreducible system of concomitants of a 
finite number of quantics is finite. 

The proof of this theorem is inductive. Let us suppose that it 
has been proved that the complete system of concomitants of any 
n quantics is finite. 

Consider a set of ?i + 1 quantics, 

y 1 > y 2 > • • • /n+i • 

The n quantics ftf, ••• fn possess, by hypothesis, a finite system 
of concomitants which may be called 8-^. The single form /„^.i 
also possesses a finite system of concomitants, which may be called 
^2- The complete system, 8, of concomitants of the n + \ quantics 
is obtained by combining 8-i^ with S^. And since the systems 
^1 and 82 are both finite and complete, it follows that the complete 
system 8 is finite. Hence if the complete system of concomitants 
of any n quantics is finite then that for any w + 1 quantics is also 
finite. But the complete system for any one or any two quantics 
is finite. Hence the complete system of concomitants for any 
finite number of quantics is finite. 

We proceed to find the complete systems, in a few of the 
simpler cases. 



160 THE ALGEBRA OF INVARIANTS [CH. VIII 

138. Linear form and quadratic. The complete system 
of concomitants of a quadratic / consists simply of the quadratic 
itself and the invariant {f,/y\ 

Thus the system S^ is 

The system S^ — of the linear form I — is simply I. 

The combined system 8 is obtained by taking all possible 
transvectants, 

But unless /3 = 0, this is equal to 

JD^if^Jyy, 

and is certainly reducible. 

Again (/«, ^)« = (/^ Py . ly-\ 

which is reducible unless y = B. 

Further {f°; l^Y contains a term 

i/jy-if-M'-'f-', 

and is reducible if S > 2. 

The system 8 then consists of 

/ A, I, (/ 0, (/ 0^. 

138 A. Linear form and any finite system. Let the finite 
system referred to be denoted by Si. The system S^ consists 
simply of the linear form /. 

Let the system ^i consist of the forms C^, 0^, ... C^ which are 
of orders Si, s^, ... Sa- 

Then' we have to consider all possible transvectants 

If /9 > 7 this transvectant contains a factor I, and is therefore 
reducible. It may then be supposed that 

yS = 7- 

Let us suppose that a^ df ; then if y>Sr the transvectant 
contains the terms 



138-139 a] simultaneous systems 161 

and is reducible. If 7 :f> 5^ it contains the term 

(6V , ly)y . Ci-' Ca"* . . . C/r-^ . . . Ox^ 
and is therefore reducible. 

Hence the system S contains the forms 

Ci, C^, ... Ca ; I; (Or, 1^)^, 

7 = 1, 2, ... Sr, 

r=l,2,...\; 
and these forms only. 

Ex. Prove that the complete system for a linear form I, and a given 
finite system of forms, consists of the linear form, the given system, and the 
forms obtained by operating with powers of 

(l ^-l ^\ 

on the members of the given system. 

139. Two quadratics. Let /j, f^ be the two quadratics. 
We have to combine the two systems *Sfi, 8^, where 8^ consists of 

and (^2 consists of 

Since D^ and Dg are invariants, they give rise to no new forms. 
Hence we have only to consider transvectants 

The only irreducible transvectants which can be obtained are 

'J\i = \f\ > J 2), 
and D,^ = {f„f^y. 

The required system is then 

Jli J2} 'J\2j ^1) -^2> -^12- 

139 A. Any number of quadratics. Consider first three 
quadratics fi, f^, f^. To obtain their simultaneous system we 
combine the system 8-^ for /i, f^ with the system 8^ for f^. 

Leaving invariants out of account we must consider all 
transvectants 

(/l"//A.2,//)^ 

G. & Y. 11 



162 THE ALGEBRA OF INVARIANTS [CH. VIII 

Since all the forms are quadratics the only irreducible trans- 
vectants obtainable are 

\f\> Js)) (/a. Jsh Wi2> Js) i 

\f\i fsfi if a, Jsf, («^i2> fzf- 

Of these {Jn,fz) is reducible, for it is the Jacobian of a Jacobian 
and another form. 

The rest are 

♦'ISj «'23> -^13> -^23> 

and another invariant which may be called 

-£^123- 

The complete system for the three quadratics is then seen 
to be 

J\'J^>j3y *^12> «'I3> •■'23> "li -^2> -^S> -^12> -^23> -^31 > -^123- 

There is only one form of a new kind, and this is an invariant. 
Hence in forming the system for four quadratics, we shall not 
meet with any new kind of concomitant. And in fact it is easy 
to see that every irreducible concomitant in the system for any 
number of quadratics belongs to one or other of the types 

/, /, D, E. 

139 B. It is easy to obtain the syzygies between the forms of 
the last paragraph. First J^^ is a Jacobian, and therefore, § 78 

2/^,=-A/2^-A/l^^-2A2/l/2 (1). 

It MFill be convenient to use the notation 

ji^^ ^x > /2 — ^a; > y 3 — Ca;', .... 

Then as in § 77 we obtain 

2 /i2 J34 = 2 {ah) aj)x . {cd) c^dx 

{acf (ad)^ a^ 

(bcY (bdy K' 

cj dx' 

= - A3/2/4 - DuAA + A4/2/3 + D^fJ, (2). 

By replacing /< by/i, a syzygy for 2Jj2Jgi is obtained. 



139 A-139 b] 

Again 
hence, as in § 77, 



SIMULTANEOUS SYSTEMS 
^123 = — (ab) (be) (ca), 



163 



2-£^123 • -C^456 — 



(ady 


(aey 


(a/y 


{bdy 


(bey 


(b/y 


(cdy 


(cey 


(c/y 


Du 


A5 


Ae 




D^ 


D^ 


n^ 




D» 


A« 


Dss 





(3). 



From this may be obtained syzygies for 

2 E\3s , 2 J&123 . ^124 . 2 £'i23 . Ey 

Similarly, the syzygy 



123"46 — 


A. 


Aa 


A 




D^ 


D^ 


A 




Du 


D^ 


fs 



(4) 



may be obtained, and other particular cases may be deduced. 



Again 



h^ 6162 bi 6a;' 

c?i=^ d^d^ di di 



= 0, 



for the last column of this determinant is a sum of multiples of 
the first three columns. 

But it has been shewn, § 77, that 

-^123 = — {oh) (be) (ca) = 



(h' 


Oitta 


ai 


V 


6162 


bi 


C^' 


CiC^ 


(^ 



Hence 

fiEn^— f^Ei^+ f^E^^— fiE^^ = (5). 

If in the above determinant the elements of the last column 
are replaced by (aey, (bey, (cey, (dey respectively, another 
syzygy is obtained, 

A5-^234 — D^E^ + D^Eyn - D^Eva =0 (6). 

11—2 



164 



THE ALGEBRA OF INVARIANTS 



[CH. VIII 



and 



In § 77, it was proved that 












{aef 


wy 


{^gy 


(ahy 






{hey 
(cey 


{b/y 

(c/y 


((^gy 


{bhy 
(chy 


= 




(dey 


(d/y 


(dgy 


(dhy 




Similarly we obtain the syzygies 






A5 


Ac 


Az 


/ 






D^ 
D^ 




n^ 

D. 


/a 


= 




D^ 


D^ 


A7 


A 






Du 


Aa 


Aa 


A 






D^ 
D^ 


A5 


D^ 
B^ 


A 
A 


= 






A 


/» 


/« 










(7) 



(8), 



(9). 



Every kind of syzygy which occurs in the irreducible system 
of concomitants for any number of quadratics has now been 
obtained. 

Ex. (i). Shew that the last three syzygies just written dowu are not 
independent of those which come before, but may be obtained from them 
on multiplying by forms of the types E and J. 

Ex. (ii). Obtain the syzygies (1), (2), (5) by means of Stroh's method. 

Ex. (iii). Obtain (4) from (3), and (6) from (5) by transvection. 

140. Quadratic and cubic. Let 4> be the quadratic and 
/ the cubic. Then we have to combine the systems of forms Si 
and S^; where ^1 contains 

and S2 contains 

/ (Afy = H, (f,H) = T, (H,Hy = A. 
All transvectants 

must be considered. Any transvectant for which either /S or 97 is 
other than zero is obviously reducible, it may then be supposed 

that 

^ = 0,77 = 0. 



139B-141] SIMULTANEOUS SYSTEMS 165 

Again both <f> and H are quadratics, hence if B is not zero 
and f > 2 the transvectant contains the reducible term 

We have then only to discuss the transvectants 

(</), u), (0, Hf, (<^%/>ro^. 

Of the transvectants 
all contain reducible terms except 

{<\>,f\ (<!>>/)'> (<^^/n («^^/)''• 

Now since, by § 91, 

those transvectants for which e > 1 are all reducible. Hence of 
the transvectants 

all are reducible except 

for (<^, T) is reducible by § 77. 

The only other irreducible transvectant is readily seen to be 

ir, fTf. 

The simultaneous system for the quadratic and the cubic / 
then consists of: 

five invariants 

D, A, {<^,H)\ (4>^f% {4>\fTf\ 

four linear covariants 

three quadratic covariants 

<f>, H, (</), H) ; 
three cubic covariants 

/. T, (<!>,/). 

141. Quadratic and any system of forms. Let the system 
of forms referred to be denoted by Si, the system S^ for the 
quadratic / consists of 



166 THE ALGEBRA OF INVARIANTS [CH. VIII 

The invariants of both systems may be left out of account as 
they produce no new forms. 

All possible transvectants 

must be discussed, where P is a product of the forms 

Giy C2, ... Cx 
of the system Si. 

If p <2i — 1, U contains the term 

and is therefore reducible. Since p cannot be greater than 2r we 
may confine ourselves to the cases 

p = 2r, 2r - 1. 

If P be a product of two factors one of which is of even order, 
then U is reducible. 

For let P = P1P2, where Pj is of order 2t, then 

(p,p,, pr 

contains the term 

(p„/r.(P2./'-')^^ 

and is in consequence reducible. 

Also if P is of order > p, and is a product of two factors, JJ is 
reducible. By what we have just proved, if one of the factors of 
P is of even order JJ is reducible ; let then, P — P^P^ where P^ is 
of order 2^i + 1 and P3 of order 2^2 + 1, then 

{PiP,, fry 

contains the term 

(p„/^r..(p„/'-'.)'-^ 

since /a < 2^ + 2<2 + 2, and therefore p — 2ti< 2t<i + 2. 

Thus JJ must always be reducible except when P consists of 
a single term C; or when P consists of a product of two terms 
Ci, Cj each of which is of odd order, their total order being p = 2r, 
so that 

U={CiCj,frr. 



141-142] SIMULTANEOUS SYSTEMS 167 

Thus the irreducible forms belong to three classes : 

(i) {c, f^r-\ • 

(ii) (c, rr, 
(iii) (c,c,-,/r, 

where Gi is of order 2^+1, and Gj of order 2r — 2^ — 1 ; this latter 
class furnishes invariants only. 

It has not been proved that all transvectants belonging to 
these three classes are irreducible ; on the contrary we proceed to 
examine a case in which certain of the transvectants thus retained 
are reducible. 

142. Let C he a, Jacobian = ((7;, Cm). 

(i) Let Ci = <f>^'-, C^ = t^^ 

/= a,^ = 6/ = . . . , G = ((jyyjr) <l>,'-'irJ^^-\ 

Then the form (G, f)^-' 

is reducible. For if 2r>2cr + 2T— 1, this transvectant vanishes; 
and if 2r <2a + 2t — 1 it contains the term 

{yjrc <" y {yjrc <=" y ... {yjrc <''-^-'' y 0a;2<r-2A-2 .,|^^2T+2\-!!r_ 

But 

{<f>y^) (<f>a) a^f^ = ^ [- {a^y<^,' + (<^^)W + {4>ciy>]r,'\ 
And hence the term written down is 

where T is a term of ((C^, G^y,f'-T'~^. 

(ii) If c^=<^,^ c^=^/r,^+^ 

then the transvectant 

(G, fryr-i 

vanishes if 2r> 2a- + 2t, and if 2r < 2o- + 2t it contains the 
reducible term 

(«^.^)(<^a)a^t^(</,6<")^..(#<"')'^ 

{fc^^^y... (slrC^'-^-^^y <j)^^-'-^-^ ^^2T+2A-2r+i^ 

We are left with the case 

2r = 2o- + 2t. 



168 THE ALGEBRA OF INVARIANTS [CH. VIII 

(iii) If Gi = 4>^^^\ Cm = ylrJ^^\ 

then the transvectant 

(c, rr-' 

vanishes if 2r >2a- + 2t + 1, and contains a reducible term if 
2r < 2o- + 2t ; but not if 2r = 2o- + 2t. 

Hence : " The transvectant (G, f"")"^-^ is reducible, if C is a 
Jacobian, except when one at least of the forms of which G is 
composed is of odd order, and the order of G itself is equal to 2r 
or 2r - 1." 

143. Quadratic and Quartic. The simultaneous system of 
irreducible concomitants when the ground-forms are a quadratic 
and a quartic may now be written down. 

The complete system for the quartic <f) is known to be 

cf>, H = (<f>,<f>y, T = {<f>,H), i = (<f>,<l>)\ j^{H,4>f. 

Since there are no forms here of odd order, there can arise 
DO irreducible concomitants belonging to the third of the three 
classes mentioned above. The simultaneous forms are 

(0,/), (<^,/n {<i>^f% (<!>>/% 

(H,f), {H,f)\ (H,f% (E,f% 

{T,fy, {T,f% {T,f^f. 

It follows from the theorem of § 142 that the forms {T, f), 
(T, f% {T, py are reducible, T being a Jacobian. 

To complete the simiiltaneous system we must take into 
account the forms which belong to the quartic and quadratic 
separately; thus in all we have 18 concomitants. 

Ex. Prove that all the forms of the complete system for the two co- 
variants J and i of a binary quintic /, considered as separate quantics, are 
irreducible when considered as concomitants of the quintic ; with the single 
exception of one invariant of degree 18 in the coefficients of /. 



CHAPTEE IX. 

HILBERT'S THEOREM. 

Hubert's Proof of Gordans Theorem. 

144. We shall now give another proof of Gordan's theorem 
that the irreducible system of invariants and covariants of any 
number of binary forms is finite. The method, which is due to 
Hilbert*, is of more general application than that of Gordan, 
inasmuch as with slight and non-essential modifications it applies 
to forms with any number of variables ; on the other hand, unlike 
Gordan's process it gives practically no information as to the 
actual determination of the finite system whose existence it 
establishes, in other words it proves that the problem always has 
a solution, while the other method, although only proving this for 
binary forms, gives much information as to the nature of the 
solution. 

In the exposition of Hilbert's proof we shall confine ourselves 
to binary forms, and to save trouble we shall deal with pure 
invariants only ; inasmuch as the complete system of invariants 
and covariants of any number of forms is really equivalent to the 
system of invariants of the set of forms obtained by adjoining an 
arbitrary linear form to the original set, the proof for invariants 
is sufficient for the most general case. Cf, § 139. 

145. The proof may conveniently be divided into two parts 
of the following purport. 

I. Proof of the fact that any invariant I of the system may 
be written in the form 

1 = -Aj/ 1 + A<il^ + . . . + -A-n^ny 
* Math. Ann. xxxvi. Story, Math. Ann. xlii. 



170 THE ALGEBRA OF INVARIANTS [CH. IX 

where /j , I^... In o.re a finite number of fixed invariants of the 
system, and the A's while not necessarily invariants are integral 
functions of the coefiicients. 

II. The application to both sides of the equation just given 
of a differential operator which leaves an invariant unaltered except 
for a numerical multiplier, and changes a term, 

Arlf 

into one of the form Jrlr, where Jr is an invariant. 

As a result of I. and II. any invariant may be obtained in 
the form 

-t i«/i + Ii^Ji + . . . + J-n^n- 

Then by applying the same argument to the J's and so on it 
follows at once that the 7's form the complete system. 

146. As a matter of fact the result I. is a particular case of a 
much more general proposition which we shall first enunciate, 
then illustrate, and finally prove. 

Theorem. If a homogeneous function of any number of variables 
be formed according to any definite laws, then, although there may 
be an infinite number of functions F satisfying the conditions laid 
down, nevertheless a finite number F^, F^, ... F^ can always be 
chosen so that any other F can be written in the form 

F=A,F, + A^F^-\-.,.-\-A,F,, 

where the A's are homogeneous integral functions of the variables 
but do not necessarily satisfy the conditions for the F's. 

Suppose for example that we have three variables x, y, z which 
we take to represent coordinates and that\^=0 represents a curve 
through the point 2/ = 0, z = ^ (this being the law according to 
which F is formed), then F may be written in the form 

2/^ + ^Q. 
as follows at once since the highest power of x must be wanting 
in the equation ; 2/ = 0, 2^ = being two curves of the system, this 
is the application of the theorem to this case. 

As another example, if the curve pass through all the vertices 
of the fundamental triangle its equation may be written 

yzP + zxQ, + xyK — 0, 



145-146] hilbert's theorem 171 

where P, Q, R are integral functions of the coordinates, and 
here yz = {) etc. are curves of the system. 

Again, we have the famous theorem that the equation of any 
curve through all the points common to ^ = and ■y^ = may be 
written 

In each of these cases it will be noted that the system of 
forms F^, Fo.,...F^ is determined; in the general case it is not 
actually determined, the essential point being that it is finite and 
that the -4's are integral functions. 

To establish the theorem in its general form we first remark 
that it is manifestly true when there is only one variable x, 
because in this case each F consists of a power of x and therefore 
all the ^'s are divisible by that which is of lowest degree ; thus 
there is only one form in the system jF,, F.^, ... Fr. 

We now assume that the theorem is true when there are n — \ 
variables and deduce that it is true when there are n variables. 

Let a?i, x.^,...Xn be the variables and first suppose that the 
system contains a form H. of order r in which the coefficient of «;„*" 
does not vanish. Then we can divide any form in which x^ occurs 
to a power equal to or greater than r hy H without introducing 
coefficients fractional in the xs, and we can continue the process 
until the remainder contains no power of x^ higher than the 
(r-l)th. 

Hence we can write any form of the system thus 

F^HP + Mxn'-' + N, 

where P is the quotient, ilf is a function of x^, x^, ... x^-t,, 
and iV^ is a function of the variables but of degree r — 2 at the 
most in x^. 

Now the functions M are formed according to definite laws if 
the jP's are, because each M is deduced from the corresponding F 
by a definite process, and as they only contain n—\ variables the 
theorem is true by hypothesis for them. 

Accordingly we can choose a finite number of ilf's, say ilfj, 
M<i,...Mk, such that any other may be written in the form 

M = B,M, + B.,M, + ...+ BJI^, 

where the B's are integral functions of x^, x.2, ...Xn-i. 



172 THE ALGEBRA OF INVARIANTS [CH. IX 

But since 
cCr:-^M=F-HP-N, Xrr'M,==F,-HP,-N„ etc., 
we have 

+ B,{F,-HP,-N,), 
or 

F==H(P-B,P,-B,P,-...-B,P,) + B,F, + ... + B,F, 

+ N-B,N,-,..-B,N,. 

Now the part of the right-hand side which does not contain 
one of the forms as a factor consists of B's and N*s and there- 
fore only contains Xn to the power r — 2 at most. Hence we may 
write 

F= HQ, + B,F, + B,F,+ ...+ 5, F, -[- if' ^»^/-^ + N^'\ 

and now il/w is a function of iCj, iCg, ... a;„_i formed according to 
definite laws and iV^''* is of order r — 3 at most in ic„. 

Thus we can write F as the sum of a finite number of terms 
each containing a form of the system for factor together with 
expressions of order 1 — 2 at most in the last variable. 

Then applying precisely the same argument to the Jlf *^*'s as we 
applied to the M's we see. that by adding a finite number of 
F's to 

we can reduce the order of the remaining portion in a?„ to r — 3. 

Proceeding in this way and adding only a finite number of 
F's at each step we can finally write F in the form 

where il/''"' only involves Xi,^^, ... Xn-i and in the nature of things 
is formed according to definite laws. . Hence applying the same 
process to the Jf' '"''s as we applied to the M's, we finally have F 
in the form 

AyF^-\-A^F^-it ...+AsFg, 

where the .^'s include H and the number s is finite. 

Consequently if the theorem be itrue for n-\ variables it is 
true for n, but it is true for one variable, therefore by induction 
it is true universally. 



146-148] hilbert's theorem 173 

We have now to remove the limitation imposed above, viz., that 
there exists a form of the system in which the coefficient of the 
highest power of ar„ is not zero. 

If there is no such form among the F's let Ft be one of the 
forms and apply to all a linear substitution 

iVr = ftriyi + 0,^2^2 + •.. + a„,y„ ; r = 1, 2, . . . w. 
Suppose that Ft (x) becomes Gt {y), then the coefficient of the 
highest power of y„ in Gt is Ft {am, ci^, • • ■ a^n). ^'"d therefore unless 
Ft is identically zero we can choose the linear substitution so that 
this coefficient is not zero*. Hence the theorem is true for the 
forms G in the variables y, and therefore by changing the variables 
back again from y to x we see that it is true for the F's. 

Q. E. D. 

147. Returning now to the consideration of invariants it is 
clear that such an expression regarded as a homogeneous function 
of the coefficients of the forms is formed according to definite laws ; 
hence, if / be any invariant of the system, we have 

where 7i, /a, . . . /,iare n fixed invariants and the A's are homogeneous 
integral functions of the coefficients but not necessarily invariants. 
As to the functions A a simple remark may be added. All that 
is asserted in the general statement of the foregoing theorem is 
that they are homogeneous in all the coefficients, but an invariant 
is homogeneous in each set of coefficients involved taken separately, 
and consequently since / and Im are homogeneous in each set of 
coefficients, J.,„ is also homogeneous in each set. That this is so 
could of course be seen in the proof of the general theorem because 
at no point of the investigation is the homogeneity disturbed. 

148. We now come to the second part of the proof, but before 
proceeding with it we must prove a necessary lemma on the 
properties of the operator D, so often used in the course of this work. 

If P be a function of ^i, ^2> Vi> V-2 which is homogeneous and of 
order X in |i, fg and homogeneous and of order yu, in rj^, rj^, then 

^m (^j)np^ ^ CoD"-'" P + C,i)'»-'»+i n (P) + . . . + C,„i)"n"» (P), 

• We assume here that unless a form vanishes identically values of the 
variables can be found for which it is not zero. It is easy to give a formal proof 
of this theorem. Cf. Weber's Algebra, Vol. i. p. 457. 



174 THE ALGEBRA OF INVARIANTS [CH. IX 

where J) = ^iV2 — ^2Vi> ^ ^^'^ ^ ^-re positive integers and the C's 
are either zero or constant. 

The result can be readily proved by induction, for we have 
„ dP ^ dP ^ d^P f J, ydP dP J. d'P \ 

and by Euler's theorem for homogeneous functions the right-hand 
side becomes 

(\ + fi + 2)P + nnp. 

Now in this result change P into D^^~^P so that X and fi are 
increased by n — 1, and we have 

O {D^P) = (X + yit + 2w) i)"-i P + DD. (i)"-i P). 

Hence 

Da (D"-'P) = {X + fi + 2n-2) D^'-^P + Dm (D^-^P), 

D^n (jy^-^P) = (\ + /I + 2w - 4) i)»-ip + ifD. (i)»-=^p), 
2)«-2n (D^P) = (\ + /i, + 4) V'-^p + D»-in (DP), 

D"-^n (DP) = (\ + /I + 2) D«-i P + D«n (P). 
By adding these results together we obtain 

a {D'P) = {w (\ + /i) + n (w + 1)1 D"-iP + D»n (P), 
which establishes the result when m = 1 for all values of n. 
Assume that the result is true for any value of m so that 
0,-^{jy^P) = CoD«-'»P + C'iD«-'«+ia (P) + . . . + C^D^n^ (P), 
then operating again with a we have 

r—m 

a"*+HD"P)= 2 c^a{D»-'«+'-a'-(P)}. 

But 

a(D"-™+'"P) 

= (w - m + r) (\ + /i + n - m + r + 1) D^-^'+'-^P + D"-*"+''a (P), 



148-149] hilbert's theorem 175 

and changing P into I1''(P) so that \, fi are each diminished by r, 
we deduce 

= {n-m+r){X+fM-r-m + n + 1) D«-'»+'-in'-(P) + X)n-m+r^r+i (p) 

when a^ is numerical. 
Thus we have 

r=l 

in other words, if the result be true for m it is true for m + 1, for 
the right-hand side is of the stipulated form. Hence by induction 
the theorem is true universally. > 

Ex. Prove that 

rX'^J i^+fi' + n-m + l)! {n-m + r)[ 

Cor. It clearly follows that if in the formal statement any 
exponent of D on the right-hand side be negative the corresponding 
coefficient is zero because only integral functions can appear in 
the process. 

149. With the aid of the above lemma the proof of Gordau's 
theorem may be easily completed. 

For the sake of convenience we shall regard Xi, x^ as the 
variables in the fundamental binary forms, despite the fact that 
in the general theorem proved above they play the role that the 
coefficients do in the remaining portion of the investigation. 

Suppose the variables are changed by the linear transformation 

X^ = 52^^! "r Vii'^2 

then an invariant I of the forms becomes (^vYL 
Further we have 

/= AJi + A0J2 + ...+ Anin 

and an invariant /,„ on the right becomes 

i^vY-'I.n. 



176 THE ALGEBRA OF INVARIANTS [CH, IX 

We now write down the identity which is the transformation of 

I = A.^li + -"2-* 2 + ... + A.filn, 

i.e. the same identity for the transformed quantics; we suppose 
a coefficient Am written in symbolical letters entirely, so that it is 
the sum of a number of terms each of which contains only factors 
of the types a^ and a,, where a is a symbol belonging to one of the 
quantics. 

If after transformation Am become Bm we have 

1 

and the equation shews at once that Bm is of order /a — /*,„ in both 
^ and 7). 

Now operate on both sides of this identity with 

_/ 8" 8" Y 

The left-hand side becomes a numerical multiple of /, viz. 
(fjb + l){fi \yl, and on the right-hand side we have 

n>^[{^r,y'^Bm]Im 

= Tm {Coi^vr-'-^Bm + C^i^vY-'^+'^B,,, + ... + G^i^y-a^Bm} 
by the lemma, since Im does not involve ^ or rj. 
But if fi — /i-m = V, then 

are all negative. 

Consequently 

t7o, (7|, ... Gy_i 
are all zero. 

Again J5,„ is of order /x, — fim = v in both ^ and tj, 

hence a-'+H^^), Xl-'+H^m), ... X>(5m) 

are all zero, and the effect of the operator on 

(^VY-Bm 

therefore reduces to a single term, namely 

G,£V{Bm). 



149-150] hilbert's theorem 177 

Now Bm is the sura of a number of terms each containing 
V factors of the type a^ and r factors of the type a,, hence by a 
fundamental theorem, 

is an invariant of the system. 

Therefore after operating with H** on both sides of the equation 
we are left with 

C 

I = lJmTm, where Jm= . ^.. Q,''{B„,) 

and is an invariant. 

Since Jm is an invariant we can express it also as the sum 
of a number of terms each containing an I^ as a factor, hence by 
continual reduction we can ultimately express / as a rational 
integral function of /j, I^,...Im, that is to say, these invariants 
constitute a complete system and, as we have seen, their number is 
finite ; Gordan's theorem is thus completely established. 

150. Syzygies between the irreducible invariants. 

Examples of relations between the members of an irreducible 
system of invariants or covariants have already been given, and 
in fact a very large number were obtained for the quintic. 

It can be deduced from Hilbert's Lemma that the system 
of syzygies is finite, that is to say if >Si = be any syzygy we can 
find a finite number of syzygies 

^, = 0, ^, = 0, ...^, = 0, 

such that 8= C^S^^- C^S^ + ... + G^S^, 

where Ci,...Cr are invariants. 

If there be such a relation, then of course all other syzygies are 
necessary consequences of 

S, = 0, S^=0,...Sr=0, 

and these constitute the finite system. 

The proof is very simple. Let /j, I.^, ... Im^e the members 
of the irreducible system of invariants, then S is a, function of 
Ii, I2, ... Im formed according to the law that it must vanish 
when for the /'s we substitute their actual values in terms of the 
coefficients. 

G. & Y. 12 



178 THE ALGEBRA OF INVARIANTS [CH. IX 

Hence we have 

S ^ GiSi + t/jOa + ... + C^Sj., 

where Si = 0, etc. are a finite number of syzygies and the C's, 
being functions of the I's, are invariants. (Cf. § 127.) 

151. Gordan's Proof of Hilbert's Lemma. 

Many versions have been given of the fundamental lemma of 
Hilbert on functions formed according to given laws, but the 
majority of them do not differ materially from the original proof 
due to Hilbert. Nevertheless Gordan has recently given a 
demonstration* which is so interesting and depends on such 
simple principles that we cannot refrain from giving an account 
of it here. We shall state it in the form of two theorems. 

Theorem I. If a simple prodtict of positive integral powers 
of n letters 

a;i*'a?2** ... «:«*", 

be formed in such a way that the exponents k^, k2, ...k^ satisfy 
certain prescribed conditions, then, although the number of products 
satisfying the conditions may be infinite, yet a finite number of them 
can be chosen so that every other is divisible by one at least of this 
finite number. 

To illustrate the scope of this theorem take the case of 
products of three letters and suppose the conditions are 

k, = 0{mod.S), 

Atg ^ "'s ^^ ' • 

The simple products satisfying the conditions are 

1*2 '*'3 > "'2 "'3 > • • • 

/*«3/jri3/)f»2 /]/% a /ft 4 ni. 9 

.*/j U/^ It's > «*'l •*'2 It's . • . > 

and it is evident that all such products are divisible by x^^a;^^ 
Again, suppose the sole condition is 

A'l A^2 ' "'3 ^ " J 

the products are 

/*» ft* ' /y«2/v»2/jf.,y». /jn 2fy« Of* tr* f ft* ft* " /y»3 
•^Ij '*'3 ) "'I > **'3 > "'1'*'3 J •*'l •*'2> J'i««'2"'3, «t'2"'3 > tX/j , . • . 

and all the products are divisible by x^ or a;,. 
* Liouville's Journal, 1900. 



150-151] hilbert's theorem 179 

Other examples could be given, but the above will suffice to 
shew the nature of the theorem which we now proceed to prove. 

If 71 = 1 the truth of the theorem is evident because all the 
products are powers of a single letter and are therefore divisible 
by that having the least exponent. 

We shall now assume that the result is true for n — 1 letters 
and prove that it is true for n letters. 

Let a^i^'a^a"* ••• ^n**" 

be a definite product F satisfying the given conditions and let 

x-^'^x^^ ... a;/" 
be a typical product K of the system. 

If K be not divisible by P one of the k's must be less than the 
corresponding a. 

Suppose that k^ < a^, then, consistently with this, k^ must have 
one of the values 

0, 1, 2, ...a^-1. 

Hence if K be not divisible by P one of a number 

«! + a-i + • • • + f*M = -^ contingencies arises, viz. 
either 

ki has one of the values 0, 1, 2, ...aj — 1, 

or k.2 has one of the values 0, 1, 2, ...a2 — 1, etc. 

Suppose that k^ = in, and that this is the pth of the possible 
cases ; then the remaining exponents k^jk^, ... Ar^i, A^^+i ...k^ satisfy 
definite conditions which are obtained by making k^ = m in the 
original conditions. 

Let Kp = x^^ x^'"^ . . . x^^ . . . x^'^ 

be a product of the system for which kj. = m and write 

Then K'p contains only n — 1 letters and the exponents satisfy 
definite conditions, and when these are satisfied the exponents of 
Kp satisfy the original conditions. Hence by hypothesis a finite 
number of products of the type K'p can be found such that every 
other such product is divisible by one at least of these. 

12—2 



180 THE ALGEBRA OF INVARIANTS [CH. IX 

Denote this finite system by 

Lx, L2, ... Lap 

80 that Kp is divisible by one at least of the Z's. 

Thus Kp = x^^K'p is divisible by one at least of the products 

which all belong to the original system of products because every 
L belongs to the subsidiary system. 

Denote these latter products by 

then in the pth of the N possible contingencies K is divisible by 

one of the products 

M '1' M <■•" M '"''^ 

■'■"■ p y •'" p J • • • •'•" p • 

Now one of these N contingencies certainly does arise when K 
is not divisible by P, and hence K must be divisible by one of the 
products 

or else by P. 

The exponents of the ilf' s all satisfy the prescribed conditions 
and they are finite in number, hence if the theorem be true for 
n — 1 letters it is true for n letters, but it is true for one letter and 
hence by induction it is true universally. 

152. Theorem II. If a system of homogeneous forms be con- 
structed according to given laws, then a finite number of definite 
form^ of the system can be chosen such that every other form of the 
system is an aggregate of terms each of which involves one of the finite 
number of forms as a factor, and the coefiUcients are integral in the 
variables. (Hilbert's Lemma.) 

Suppose in fact that Xi,X2, ...x^ are the variables and that (f> is 
a typical form of the system. Now construct an auxiliary system 
of functions ri of the same variables according to the law that a 
function is an rj function when it can be written in the form 

17 = XA<f), 

the A'a being integral functions of the variables which make the 



151-152] hilbert's theorem 181 

right-hand side homogeneous, but otherwise unrestricted except 
that the number of terms on the right-hand side must be finite. 

The class of functions r) is infinitely more comprehensive 
than the class <^, and it possesses the important property that a 
function of the form ^Brj which is homogeneous in the variables 
is also an 17 function. 

Now in examining the functions 7; we arrange the terms of one 
of them of order r in such a way that x^ comes first and, generally, 
a term 

comes before a term 

when the first of the quantities 

which does not vanish is positive. 

In such a case we say that the term S is simpler than the 
term T and T is more complex than S, so that any function rj is 
arranged with its terms in ascending order of complexity. Now 
the functions rj being formed according to fixed laws, their first 
terms satisfy given conditions relating to the exponents, and 
hence by Theorem T. a finite number of 77 's, say 'r}i,7)2, ... rj^, can be 
chosen such that the first term of any other rj is divisible by the 
first term of one of these. 

Take any function 77 of the auxiliary system, and suppose its 
first term is divisible by the first term of t/^,. and that P^ is the 
quotient. 

Then v — PiVmi is an rj function with a more complex first 
term than rj because ij and Pi'>7m, have the same first term; if we 
denote this new function by t] '^* we have 

Next, if the first term of 7/ 'i' be divisible by that of rj^^ we 
have 

», (1) _ P „ 4. «, (2) 

and the first term of t] <^' is more complex than that of 77 '^>. 
Continuing this process of reduction we find 
7; ('■-1' = PrVmr + V ^^^ and so on. 



182 THE ALGEBRA OF INVARIANTS [CH. IX 

Now the first terms of 

are in ascending order of complexity, and hence the time must 
come when there is no 77 function of the same order as rj with 
a more complex first term than rj ""> ; in that case we have 

Hence 

where the 77 's on the right-hand side are all members of the finite 
system 77i,?72,...77p. 

Now the 7} system includes all the <^'s, moreover each 7; 
contains only a finite number of ^'s and hence every ^ can be 
expressed in the form 

where <^i, (f>2,...<f>r are the <^'s contained in the expression for 
Vi} V2> •'• Vp ai^d the A's are integral functions of the variables. 

153. Remark. If all the conditions satisfied by k^, k^,.., k^ 
in Theorem I. be linear homogeneous equations, then the theorem 
establishes the existence of a finite number of solutions by means 
of which any other solution can be reduced. The difference of 
two solutions being now a solution, it follows that by continual 
reduction we can express all solutions of the linear equations in 
terms of a finite number — this is the result otherwise proved 
in § 97. 

* The ij's on the right being members of a finite system are finite in number ; 
hence even though the number of steps in the reduction be infinite, there can 
only be a finite number of terms on the right-hand side. The same ^ may of course 
occur in more than one term, but in that case we should add all such terms 
together. 



CHAPTER X. 

THE GEOMETRICAL INTERPRETATION OF BINARY FORMS. 

154. Given two points of reference A, B on any straight 
line, the position of any other point P may be determined by the 

AP 

value of the ratio ^-^ of the 

A P B "i> 

distances of P from A and B. A 
convention as regards sign is necessary to complete the definition ; 
it is convenient to regard the ratio as positive if P lie between A 
and B, otherwise as negative. 

When a binary form of order n is equated to zero, the ratio — 

may have any one of n values. These determine n points on the 

AP 

straight line AB, such that the ratio ^^ for each of these points 

is equal to one of the roots of the equation for — . The coordi- 

nates {x^, x^ then define the position of P on the straight line by 
means of the equation 

AP ^x, 

PB ~ x^' 

It is found advisable, as will be evident immediately, to define 
the position of the point P whose coordinates are (i^i, ^2) t>y 
means of the equation 

AP_.x, 

PB~^xl' 

where A, is a fixed numerical multiplier. 

A further convention will be useful, viz. the positive direction 
of measurement from A is towards B and that from B is 
towards A. 



184 THE ALGEBRA OF INVARIANTS [CH. X 

To find the distance between {x^ , x^ and {yi , y^), in terms of 
the length I of AB. 

Denoting the two points by P and Q, we have 

AP^PB^ AB _ I 

\Xi x^ Xxi + x^ \Xi + X2 ' 

and similarly 

QB^ I 

2/2 ^yi + y-z' 

Hence 

pq - {PB - QB) - ^_-^^___^^^__^ 

XI (yx) 



(^2/1 + ^2) (5^1 + 573) 



155. Let us consider the effect of a change in the points of 
reference. Let the new points of reference A', B in terms of the 
original system of coordinates be (^1, ^i), (171, tj^); if the new 
multiplier be fi and {Xi, X^) be the new coordinates of P, then 





X, A'P 
X,~^PB" 




^^^1 + ^2' (v^) 



Hence 

The change in coordinates is thus equivalent to the linear 
transformation 

Xi = pi (x^), 

where -=/'^l4f- 

156. A linear transformation 

X2 = ^2X1 + 7)2X2, 
may be regarded geometrically from two different points of view : 

(i) As changing the points of reference and the constant 
multiplier, but leaving the other points on the straight line in 
their original position. 

(ii) If the points of reference are regarded as fixed, the 



154-156] GEOMETRICAL INTERPRETATION OF BINARY FORMS 185 

transformation alters the positions of all the points defined by the 
algebraic forms under discussion. 

Consider the first of these points of view. When x^ = 0, 
the point P(oei, x^ coincides with one of the points of reference 
A. Similarly, if a^j = 0, P coincides with B. Hence to find the 
new points of reference in the original system of coordinates, it is 
only necessary to write X-^ — 0, and X2 = 0. We obtain them at 
once as (^1, t]^ and (fi, fg)- 

The distances of P from these new points of reference A\ 
B' are 

\W. ^?^^- .=u (^'^>^^ 



{\x^ + x^ {Xtji + 7/2) (A^i + ^2) (^^1 + %) ' 

and XI ^ ^.-— .. = X/ ^^'^^^^ 



(}U;i + x^) (X|i + I2) (\^i + x^) (Xf 1 + ^2) ' 

The ratio of these distances is 

A'P \^x + ^2 ^1 



PB Xvi + V^'^ 



Hence the new multiplier is -^ — —; and the coordinates 

{Xi , X2) define the same point as that defined by the coordinates 
(xi, X2). It should be observed that the sign of the expression 

— ^^ -^ . ^ is positive if X lie between A' and B', otherwise it is 

negative. 

Ex. (i). Shew that by properly choosing quantities oj, ag the distance 
between the points (^1, x^), i^y, y^ maybe written ~^ . And that in 

this case the constant multiplier after transformation becomes ,— ^ . 

Ans. a^=-^ -, a^^sj J. 

Ex. (ii). The point (aj, a^ of the last example is the point at infinity on 
the range. 

When an invariant of a binary quantic is zero, there exists 
some relation between its roots which is unaffected by any linear 
transformation. Hence when the binary quantic is regarded 
as the analytical expression of n points on a range, the vanishing 



186 THE ALGEBRA OF INVARIANTS [CH. X 

of an invariant is the condition that there may be some definite 
geometrical relation between the points, independent of the points 
of reference and of the constant multiplier. 

For example if two of the points coincide an invariant — the 
discriminant — vanishes. Again, as will be shewn later, if four 
points on a straight line which form a harmonic range are 
represented analytically by a quartic, then the invariant j of that 
quartic is zero. 

157. In the second point of view stated in the last paragraph, 
the points of reference and multiplier are regarded as fixed, the 
point P takes a new position P' given by the coordinates 

Let the points Q, R, S, viz. (y^, y^, (z^, z^), (Wj, w^) become 
Q', R\ S\ 

Then PQ = xr, ^^ — , 

and (ya^) = (^v)(YX), 

Hence PQ^BS _{y^wz) 

PS.EQ~{ww)(yz) 

_ (rX)iWZ )_ P'Q'.R'S ' 
~(WX)iYZ)~' P'S'.KQ'' 

The expression p^—pTs is called the cross (or anharmonic) ratio 

of the four points P, Q, R, S ; it is usually denoted by {PQRS] * 
The result just proved may be written 

{PQRS} = {P'Q'R'Sl 

* By rearranging the letters P, Q, R, S we obtain 24 such cross-ratios. It is 
easy to see, however, that only six of these are different. Then if X, fi, v are written . 
for the three products PQ . RS, PR . SQ, PS . QR, the six different cross-ratios are 

\ /l V fl u \ 

fl P \ \ fl V 

Since the four points are collinear, 

PQ.RS + PR.SQ + PS.QR = 
or X-f-^ + i' = 0; 

by means of this relation all six cross-ratios of four points may be expressed in 
terms of one of them. This mode of presenting the subject is due to Mr R. R. 
Webb. 



156-157] GEOMETRICAL INTERPRETATION OF BINARY FORMS 187 

That is, the cross-ratio of four points on the range is unaltered by 
any linear transformation. Hence the transformed range is homo- 
graphic with the original range. 

Further any range homographic with the original range may 
be obtained from it by a linear transformation. To prove this, it 
is only necessary to prove that the coefficients of transformation 
may be so chosen that three non-coincident points P, Q, R of the 
original range are changed to any three non-coincident points 
P', Q', R chosen at random on the straight line. For when 
P', Q, R' are known, the point S' of the transformed range 
corresponding to S is given by 

{P'Q'R'S'] = {PQRS]*. 

X Y Z 

Let us suppose then that the values of the ratios -^ ,^ , -^ 

are given. 
Then x. 



and 



or 



Similarly 

and ^2 7- . -' - ^1 ^ + '?2-^ - '/i = 0. 

■"2 ■*2 ^2 ^2 

We have three equations to determine the ratios of the 
coe ffi cients f 1 , |^2 , "^i , Vi • 

These ratios are thus determined uniquely. 

Hence a range of points on a straight line may be transformed 
into any other range homographic with itself by a linear transfor- 
mation. 

If an invariant of a binary quantic representing a range of n 
points is zero, these points must possess some special property, 

* This must give a unique position for S' since it is equivalent to a linear 
relation between its coordinates. 



^1 




+ ..) 


X^, x^=\P., 


X, 

'X2 


+ '72 


X, 






^ 




+ '/! 














SC^ 




+ »72 










1. 


X, 

'x. 




f4:+ 


•'72 


% = 


= 0. 






^2 




2/2 


4:+ 


« 2/1 

y2 


'71 = 


= 







188 THE ALGEBRA OF INVARIANTS [CH. X 

which is also a property of all ranges homographic with the 
original range. Such a property is said to be projective, and 
thus the vanishing of an invariant must be the condition for the 
existence of some projective property of the points which the 
quantic represents. 

Conversely, if a system of n points on a straight line possesses 
some projective property, there will exist a corresponding 
analytical relation between the coefficients of the quantic repre- 
sented by these points, which is unaltered by any linear 
transformation. It does not necessarily follow that the condition 
is represented by the vanishing of an invariant; it soijietimes 
happens that a projective property necessitates the vanishing of 
all the coefficients of a covariant. 

Again a covariant of a quantic will define a certain number of 
points on the straight line. These points are related to the 
original points of the range in the same way as their homo- 
logues are related to the homologues of the original points on 
a homographic range. It is usual to denote this by saying 
that the points are projectively related to the points of the 
original range. 

158. A binary form is homogeneous in two variables, we are 
then — in such forms — only concerned with the ratio of the 
variables. Let /{x^, x^) be any binary form of order n, then the 
equation 

f{x„x.,) = {) 

defines n values of the ratio — . Hence in any geometrical figure 

x^ 

in which the geometric element is completely defined in position 

by a single parameter, the iorici f {x^, x^) may be considered as 

defining n of these elements. For example a point which lies on 

a unicursal curve is such an element, li x, y, z are its Cartesian 

coordinates, it is well known that we may express x, y, z bs, 

rational algebraic functions of a single parameter. Again the 

tangent to a fixed unicursal curve may be taken to be the element. 

Or else the element might be the osculating plane of a twisted 

unicursal curve. 

Now the two simplest figures of the kind, are a range of 
points on a fixed straight line, and a pencil of straight lines 



157-159] GEOMETRICAL INTERPRETATION OF BINARY FORMS 189 

passing through a fixed point and lying in a fixed plane. We 
may deduce the properties of the latter from the former. For if 
any straight line be drawn to cut the pencil there is a one-to-one 
correspondence between the points on the range thus formed and 
the rays of the pencil. In fact any ray of the pencil may be 
defined by the coordinates (x^, x^) of the point in which it 
intersects the straight line. With this definition it appears that 
everything that has been said for the range applies equally well 
to the pencil. 

159. A binary form may be expressed as a product of n linear 
factors. A covariant of the binary form is necessarily a covariant 
of the system of linear forms of which it is a product. Thus let 

aa;" = (a;a;W)(^^"')---(^«'"') (I). 

X '^* X "^' 

where -^ , ^r, , ... are the roots of the equation aJ^ = 0. 
Any invariant may be written in the form 

since it is an invariant of the linear forms (a;^*"), (ara;''") .... 

The coefficients of the quantic are given in terms of the roots 
by equating the different powers of x in (I). Two things are at 
once apparent. 

(i) The coefficients are symmetric functions of the quantities 

a;W, a?<=" .... 

(ii) The coefficients are functions homogeneous and linear in 
each of the n sets of variables a^i''', x^'^^ ; a;i<^', x./^^;... a?!*"', a^g*"'. 

It follows that any function of the coefficients must, when ex- 
pressed in terms of the quantities x^^\ a;'^' ..., be symmetrical in 
them. And further such a function must be homogeneous and 
of the same order in each of the sets of variables Xi^^\ x^^^^;... 

^ («) ^ (n) 

Hence in the expression for an invariant (II) it is necessary 
that 

where ^ is a quantity which is the same for each terra of the sum 
representing the invariant. 



190 THE ALGEBRA OF INVARIANTS [CH. X 

Let T be any one term of this sura, then let 

Then iSi2 + /3,3+...+^x„ = 0, 

Ai + /923+...+/32« = 0. 

We are going to prove that ^ is a function of anharmonic 

ratios of the roots. It will be assumed that when the number of 
quantities 3o^^\ a?® ... is less than n, then the term 

where 2/Si^ = 0, 1^^ = 0, 

r r ' 

may be expressed as a function of the anharmonic ratios. 
Now the ratio 

(^(1,^(2,^,4,^,0} =l^^)i£^. 

Hence 

^ ^~^ ^ ''•(a;Wa;<=»>)'{ic<«ar<2>a;Wa;<*)}* 

On replacing (aj'^'a;'") wherever possible by the value just 
found 

becomes 

where P is a function of anharmonic ratios, and Q is of the form 

where S/Sar = 0, SySsr = 0. 

r r 

The theorem has been assumed true for Q, hence with this 

assumption it is true when there are n quantities a;<'>, a;*^', If 

there are only three quantities, then 

y3i2 + ^i3 = 0, 

/^Sl + ^32 = 0, 

and J3n = = ^^ = ^.^,. 



159-161] GEOMETRICAL INTERPRETATION OF BINARY FORMS 191 

Hence it is true when there are 4 quantities x, and therefore 
also when there are 5, and so on universally. Thus, any invariant 
of a binary form is the numerator of a rational function of the 
anharmonic ratios of the 7'oots. If the invariant contains only 
one term, there is an apparent exception. The invariant equated 
to zero then represents the condition for the equality of a pair of 
roots, it can only be the discriminant. 

160. So far our remarks have been confined to the case of a 
single quantic ; a slight alteration in the wording of the previous 
paragraphs is all that is necessary to make them applicable to 
any system of binary forms. Each binary form of a system is 
geometrically represented by a set of points on a range, or of 
rays of a pencil. Points belonging to the same quantic must be 
regarded as indistinguishable from one another. Thus if we have 
a set of n points on a straight line, we may regard them as given 
by a single quantic of order n ; by two quantics of orders r and 
n — r respectively, or even by n separate linear forms. 

Now let two of these points coincide ; then, if the n points are 
regarded as a single quantic, the discriminant is zero ; but there 
is nothing to tell us which roots coincide. We may regard the 
n points as two quantics, in this case either the discriminant of 
one of the quantics or else the resultant of the two is zero. 

161. We shall now discuss the geometrical representation of 
the invariants and covariants of the binary forms of lowest order. 

A quadratic has only one invariant, this vanishes when the 
points representing the quadratic are coincident; it is the dis- 
criminant. 

A pair of quadratics aj^, h^ have a simultaneous invariant 
{ah)\ 

Then if a;<i>, x^"^ are the roots of a^ and 3/^, y*^) those of h^, 
(abf = (a2/<i>) (ay^'^) = ^ [(a;("y<«) (a;'^'^*^)) + (a;<«y(«) (ar'^'y «)]• 
If {aby=(), it follows that 

(x^^)y(^))(xi^)yW)~ 
Hence the pair of points 3/*^', 1/'*' is harmonic with the pair 



192 THE ALGEBRA OF INVARIANTS [CH. X 

The quadratics have a covariant, their Jacobian, ^ = {ab)axbx. 
Now (a^)^ = 0, (6^)'^ = 0, 

hence ^ represents the pair of points which is at the same time 
harmonic with a^^ and with bj?. In other words ^ represents the 
pair of double points of the involution defined by the two pairs 
a^^ bj". 

The discriminant of ^ is 

{^yy = i {{aaj (bbj - [{abm ; 

if this is zero, ^ is a perfect square ; the double points of the 
involution coincide. Hence, as may be verified either geometrically 
or algebraically, one of each of the pairs a^^, b^ coincides with the 
point represented by ^, and the other two points may be anywhere 
on the range. Thus a^, b^ have in this case a common point ; 
hence (^^')^ may be taken to be the resultant of the two 
quadratics. If there are more than two quadratics, there is only 
one more type of concomitant to be discussed ; viz. the invariant 

(a6) (6c) {ca). 
This is equal to — (^, c^y. 

If this is zero then the pair of points ^ is harmonic with the 

pair Cx- Hence 

(a6)(6c)(ca) = 

represents the condition that the three pairs of points a^, b^, c^ 
should be pairs in involution. 

To find the anharmonic ratio of the four points defined by 
O'x, bj^, we have 

(a;Wy(«) (a;®2/»') + (a;(i'y<«») (a;(2)y<i') = {ahf, 

= v^(a«7.V(667. 
Hence 

2 (a;(i'y<«) (a;»»y W) = {aby + ^{aa'f.{bbj, 

2 (a?»'y ('^O (a;<«y <«) = {ahf - ^/{aa')\{bb')\ 
and therefore 



^ (aby + ^(aay.(bby 
{aby - ^{aaJAbUy' 



161-163] GEOMETRICAL INTERPRETATION OF BINARY FORMS 193 . 

Denoting the anharnionic ratio by p, and squaring, this equation 
becomes 

D'^(p-iy=D.D"{p + iy-, 

or P'-^P j)'2_])j)- + ^ = ^' 

where i) = (aa')-, D' ={aby, D" = {bhy. 

The two values of p correspond to the two anharmonic ratios 
ja;(i)y(i)a;<2>2/<-'»} and {aj'^'y^^iai^/Wj. 

If ^ = 0, then the two quadratics are such that one is a 
multiple of the other. This is merely a particular case of the 
general property of the Jaeobian ; it is not necessary to do more 
than mention it here. 

162. When a range possesses geometrical peculiarities which 
are unaffected by projection, there exist analytical relations of 
an invariant nature among the coefficients of the corresponding 
binary form ; but it must not be supposed that these relations can 
always be expressed in terms of the pure invariants of the form. 
If there is only one such relation 

which expresses the necessary and sufficient condition that the 
range may possess a certain projective property, then it will be 
found that A is an invariant, for it is unaltered by linear trans- 
formation. On the other hand, when the condition is expressed 
by a set of algebraical relations 

^=0, 5 = 0,... 

A,B,... will not, in general, be invariants. Thus the condition 
that a binary form of order n may be a perfect nth. power is that 
all the coefficients of its Hessian vanish*. 

163. The Cubic. Any three coUinear non-coincident points 
can be projected into any other three collinear non-coincident 
points ; it is not to be expected then, that the geometry of a 
binary cubic will be of much interest from a projective point of 
view. But in respect of the associated points furnished by covariants, 
the geometry of the binary cubic is highly interesting. 

* See later, Chap. xi. 
G. & Y. 13 



194 



THE ALGEBRA OF INVARIANTS 



[CH. X 



The single invariant A is its discriminant and 

A = 

is the necessary and sufficient condition that two of the three 
points represented by the cubic should coincide. 

If all three points coincide, then A, H and t are all identically 
zero, as may be easily verified ; but 

represents the necessary and sufficient condition*. 
164. Let us consider the pencil 

where k and \ are new constants which determine the particular 
members of the pencil of cubics. Theny^ ^ represents three points, 
which are called a triad ; by varying k and \ a pencil of triads is 
obtained. The covariants of f^^^^ will be denoted by the symbols 
■ffic,A> ^(c,\> ^K,\' These may be at once calculated; the following 
table, most of the results of which are proved in Chap, v., will be 
found useful for this purpose. 



Index 


Transvectant 


{f,.f) 


ifrS) 


(/,0 


{H,H) 


{H,t) 


{t, t) 


1 





t 


-^m 





W 





2 


H 








A 





iAzr 


3 







A 










The fundamental forms, it will be remembered, are connected 
by one syzygy, 

t' = -^[H' + A/^}. 

To obtain H^x we have 

E^^x = {Kf-h\t, Kf+\ty 

= fc' (f,fr + 2«x (/ ty + x^ (t, ty 

= (/c" + i A\^) H. 

• Chap. XI. 



163-165] GEOMETRICAL INTERPRETATION OF BINARY FORMS 195 
Hence, if we use the notation 



H.,, = %H. 



In the same way, we obtain 

= © (/c« - ^ AX/) 






And lastly ^k.\= @'^- 

It is worth noticing that if we introduce the arguments /c, \ 
of (s) as suffixes, thus 

the syzygy may be written 

165. Consider the relation 

if any pair of the three forms 

/, H, t 

have a common factor, then all three must have this factor. Let 
us suppose that such a factor exists, and let us change the variables 
so that the common factor is x.2. Then /is of the form 

(0, Oi, tta, tts^^i. «2)^ 

and the coefficient of x^ in H '\s —Oi; hence if x^ is a factor of 
H, we must have a^ = 0. This means that / has a double factor, 

and therefore 

A = 0. 
The syzygy then becomes 

whence it is easy to deduce that ^ is a perfect square and t a 
perfect cube. In fact if 

/= K% 

then H=A^,t = ^/^i:M^ = 0, 

where ^ = - I [(?<?)?• 

13—2 



196 THE ALGEBRA OF INVARIANTS [CH. X 

166. It will now be supposed that f, H, t have no common 
factor. The syzygy may be written 

hence if ^, 77 are the factors of H, so determined that 

we may take 2p =\t-\- /y -^ ) ' 

and 2v^=(t-f^-^^y 



and therefore 



Hence also 






It is at once apparent that the only members of the pencil 
which possess double factors are ^^ and 7]\ 

Let Pi, P2, P3 be the three points determined by any one 
Tnember of the pencil, and A, B the two points determined by 
the Hessian (which are the same for every member of the pencil). 
Then if w be a cube root of unity, the points Pj, PzyP^ correspond 
to linear forms 

f — at), ^ — (oarj, ^ — o)-ar). 
The ranges 

A, A, P„ P,, B 

A, P,, P„ P„ B 

A, P3, P„ P„ B 

are projective. A set of points such as Pj, 2\, P, are said to be 
cyclically projective. 



166-167] GEOMETRICAL INTERPRETATION OF BINARY FORMS 197 

The simplest way to find the anharmonic ratio [APiP^Ps] is 
to transform the variables to ^, rj. This transformation may be 
regarded as merely changing the points of reference, and the 
constant multiplier. Then 

Hence the range formed by a triad and one of its Hessian 
points is equian harmonic. The six distinct cross ratios of such a 
range are each — to or — co^. 

167. The Quartic. Just as in the case of the cubic we 
considered a pencil ot cubics instead of the single one, so here we 
shall find it convenient to consider the pencil 

instead of the single quartic f. Each member of the pencil will 
define four points, one of these points may be chosen at will on 
the range considered, but when this is done the ratio k :\ is 
fixed, and the remaining three points are uniquely determined. 
The calculation of the covariants of f^^x. presents no serious 
difficulty. For convenience a table of the transvectants of the 
quartic is appended ; most of the calculations were effected in 
Chap. V. 



Index 


Transvectant 


(/,/) 


if,H) iH,H) 1 (f,t) 

1 


{H,t) 


{t,t) 


1 
2 
3 





t 





i,iP-\m 


IfUf-iS) 





ff 


¥f 


kJf-¥S 








yHf-i^im-^\i-^p 











kUf-iS) 


ii^f-\js: 





4 


i 


j 


¥' 












The only transvectants which are not contained in this 
table are 



198 THE ALGEBRA OF INVARIANTS [CH. X 

The syzygy between the forms is 

In connection with the system for/«_x the expression 

or more briefly H, will be found of great importance. We observe 
at once that the syzygy may be written 

Now 

_ / an an\ 



K.K = kH + 2k\j + \4 i" = - 3 (n, ny 

= — SHa, 
where fl is regarded as a binary cubic in k and \. 
Lastly 

r .an . / an ^dn\ , .^^ ani 
•^-^ = n"''9^^/H"a.-^axj + **^ax} 

168. The invariant of the cubic CI is 

This, as we proceed to shew, may be taken to be the discriminant 
of the quartic. The discriminant is the condition that the equation 

/=0 
may have a pair of equal roots. 

It is an invariant ; for if the range represented by / be 
projected, the pair of coincident points project into a pair of 
coincident points. Further it is well known to be of degree 



167-168] GEOMETRICAL INTERPRETATION OF BINARY FORMS 199 

2 (w — 1) for the quantic of order n ; hence it is of degree 6 for the 
quartic. It is then a linear function of i^ and j^. 

Let us suppose that f has a double linear factor ax, the 
remaining quadratic factor being p^^, and then find what relation 
exists between i and j for the quartic 

la the first place 

H={aa?.px\ a^J 

= \ {{aaypj" + {payaj" + 4 (aa) (pa) a^Px] ««' 

= i{S{aaypx' + S(payoix'-2{apyax']ax' (I). 

But 

{aafax' = (/, ax^f = i {apf . ax\ 

(par ax' = (/ px'Y = I [{oLpfp.' + 3 {p, pyax% 

Therefore 

H^-^{apff+l{p,pra\ 
Similarly 

i = (aa)- (apY 

=((oiayax\ px'y=h[(^pyf> 
i=(/. Hy=-i{^pr.i 

Therefore 

i^ - 6/ = 0. 

Hence An may be taken to be the discriminant of the quartic /*. 

From the above form for H, it is evident that H contains the 
factor ax twice over ; hence if / contains a repeated linear factor 
then every form of the pencil /c/4- ^H contains the same repeated 
factor. This leads us to expect that the discriminant oi Kf+\H 
is a multiple of the discriminant of /. This is so ; for from the 
syzygy for the cubic we obtain 

-^n«-2^n^ = An.n^ 

or (i,x^ - 6 j,, x') = ^' (*■' - Qj% 

It is easy to obtain in the same way the condition that / may 
have two pairs of repeated factors. For writing as before 

/= CLx' • Px^, 

we obtain 

t = {f, H) = iip,pY(f, a)a,^ 



200 THE ALGEBRA OF INVARIANTS [CH. X 

Now if/ contains two pairs of repeated factors, (p, pf must 
vanish, for p,^ is a perfect square ; hence in this case 

t = Q. 

This is the necessary and sufficient condition, for if it is satisfied 

{t, ty=iQi^-f) = o, 

and we are at liberty to assume that f has one repeated factor ; 
then using the relation 

we see that either 

{p,py==o, 

in which case ^ is a perfect square ; or 

(/ «) = 0, 
in which case 

/= «.^ 
This furnishes another illustration of the remarks in § 162. 

Ex. (i). Shew that the necessary and sufficient condition that /may have 
a three times repeated factor is 

i=0, j=0. 

Ex. (ii). There are in general three difiFereut members of the pencil fg^x 
which are perfect squares. 

They are given by solving the equation 

£2=0. 

169. As has been already pointed out, the syzygy for the 
quartic may be written 

If Wi, mj, mg be the roots of the cubic 

n = o, 

then fl,t, ,\ = ('c — niiX) (« — m^X) (k — wigX). 

Hence also 

2t' {H + mj) (H + m,f) (H+mJ). 

If H and / have a common factor, by transformation this may 
be made x^. Then / is of the form 

(0, Ui, a^, a,, tti^Xi, x^y. 



168-169] GEOMETRICAL INTERPRETATION OF BINARY FORMS 201 

In order that a^a may be a factor of H, we must have 

Hence x<^ is a factor of/; and therefore 

An = 0. 

Exchiding this exceptional case, it is evident that no pair of 
the expressions 

H + Wi/, H + niof, H + m-if 

have a common factor (mj, m.^, m^ are distinct) for An is the 
discriminant of II. Hence the above relation shews that each 
of the expressions H + 'mf must be a perfect square — since f^ is a 
perfect square. 

Let 

H + m^f= — 2-v/r2 
H-¥m,f^-2x\ 
where <^, -v/r, p^; are binary quadratics. 

Then 

t = 2<^^^x• 
As an example it may be verified that for the quartic 

H= 2ax^* + 2 ( 1 - 3a2) Xj^x^^ + 2ax.^, 
Q = K3-(3a2 + l)«X2-2a(l-a2)X3, 

and that the roots of 

Q = 

are a— 1, a+ 1, -2a ; which are identical with the values of m which make 

H+7nf 
a perfect square. 

Now 

(7/ + nil/, H + tiiof) = (nil — ^^2) t 

= (-2<^^ -2>|r2) = 4(</,, A|r)<^^; 

putting in the value of t we obtain 

2(<^, -(/r) = (m, - ma) %. 
Similarly 

2(^, X) = (m2-m3)«^, 

2(x, <^) = (w3-mi)'»/r. 





2 




(rrh- 


- iri^) (wii ■ 


-TOg) 




2 




(ma- 


- ms) (to., ■ 


-TOi) 




2 











(W-j- 


- nh) (wig ■ 


-TOi) 



202 THE ALGEBRA OF INVARIANTS [CH. X 

Now from the expression for the Jacobian of a Jacobian we 
obtain 

Hence by repeated use of this formula 

={<t>xyf-(<f>n-x^ 

X = (4><f>) X-iM <!>' 

X = ii^^f X - i^xT '^' 
= ('^<f>T X - ("^xT </>' 

=ix-f)''t>-ix<t>)'f- 

Since cji, >/r, ^ have no common factors these equations give 
the following six relations 

{<}><l>y = _ |(mj - TO,) (tox - TO3), (fxy = 0, 
(i^y = - ^ (ma - TO3) (TO2 - mi), {x<l>y = 0, 
(XXY = - i (^^3 - m,) (m^ - TOa), ((fyyjry = 0. 
The remaining invariant of these three quadratics is 
((}i^|r) (yfrx) (%</>) = - ((<^V^) <f>xirx, X^xf 

= - ^ (?ni - TOa) . ixxf = - i (Wli - Wla) (Wla - Wis) (wis - »Wi). 

170. By means of the equations 

5'^-TOl/=-2<^^ 

H + mJ=-2x' (II), 

the quartic /, or more generally Kf+ \H, may be separated into 
quadratic factors. 



169-170] GEOMETKICAL INTERPRETATION OF BINARY FORMS 203 

2 

and Kf+\H= — {(k - \m,) %^ - (/c - Xm,) -f'] 



mj — Wg 



(V/e — Xm^ % + V/c — Xwis i/r) (V« — Xwg % — V/c — Xmg a^). 



The second transvectant of either of these quadratic factors 
with (f) is zero. Hence the two points determined by (f> are the 
harmonic conjugates of the two pairs of points represented by the 
above quadratic factors of Kf+ \H. Now we have only used the 
last two of equations (II) to find the quadratic factors of /c/'+ XH. 
Any pair of the three equations might be taken. The three 
results represent the three ways into which the quartic Kf+XH 
may be separated into quadratic factors. Then the three quadratic 
factors of t are the three pairs of points harmonically conjugate 
with respect to the four points Kf+XH, when divided into two 
pairs of points. 

Now (cf>r}ry = 0, {^Jrxy = 0, (x<^>^ = 0, 

hence the pair of points (f> is harmonically conjugate with respect 
to each of the pairs -v/r and x- If the points Kf+ Xll are divided 
into two pairs in any way, these pairs determine an involution, 
one of the quadratic factors of t represents the pair of double 
points of the involution. The other two quadratic factors of t 
represent pairs of points belonging to the involution. 

Now the points determined by t are independent of k and X, 
hence the pencil /c/"+ XH represents sets of four points such that 
when any set is separated into two pairs of points, these are pairs 
of one of three fixed involutions. 

The quartic / is arbitrary, it may represent any four points. 
Hence the pairs of double points of the three involutions deter- 
mined by four points on a line are harmonically conjugate two 
and two. 



204 THE ALGEBRA OF INVARIANTS [CH. X 

171. To determine the anharmonic ratio p of the four points /. 
We have obtained the quadratic factors ofy, one pair is 

X-'f' X + ^- 
The anharmonic ratio of the four points determined by a pair 
of quadratics has been obtained in § 161, as a root of the equation 

In our case 

= - ^ (mg - mi) (mg - m^ - ^ {m^ - w^ (m^ - m^) 

= -^(nis- m,y = D". 

D' = -^ (ms — mi) (mg - m^) + \ {m^ — m^) (m^ — m^). 

Hence . _ K- ^0^ + (^.- ^^O^ ^ ^ ^ ^ 
(mg - mj (ma — mj 

As there are six different values for the anharmonic ratio of 
four points, a sextic for p is to be expected. This will be obtained 
by multiplying together the three equations similar to the above. 

It will be more convenient to write these equations in the form 

(mg - mO (7^2 — mi) '^ 
Now mi, ma, mg are the roots of the cubic 

a;3 - ^ /«;X2 _ I X3 = 0. 

The discriminant of this is 

(mi - m^"^ {irru, - m^y (mg - mi)'^ = \ (i^ - 6f) = A, 

the exact expression is most quickly obtained by using the 
equation of the squared differences of the cubic. 

The equation whose roots are mi — m^, m^ — mg, mg — m.i is 
obtained thus 

S (nil — ^2) = 0, 

2 (mg — mi) (mi — m^) = — '^m^ + Smamg 
= SSmamg = — 32, 
(mj — ma) (mg — mg) (wig — mi) = VA. 



171] GEOMETRICAL INTERPRETATION OF BINARY FORMS 

The equation is ]f — '^ny~ "^ ^ — ^■ 

The equation whose roots are (7/ii — miY, ... is 



205 



or 



or 



But 

Hence 

(1-6 

Therefore 



(z - VA)' = 27 ^ ^. 

" o 

(m., - W3V = - V A " 

P 



aM- 



p'-^P + l _iY = _ 27 ^' P'-^P + ^ ^,\ 

8 



1 + 



P J op 

p^-2p + lV*_p2-2p+2 27 
p * 4 



I 



24(p2-p + l)=' 



j2 4(p^-p + l)^-27pnp-l)^ 

{p + inp-2y{2p-iy 

* When the quartic is not treated by the symbolical method it is usual to 
define the invariants as follows : — the quartic itself is 

f=(a, b, c, d, e\xi, Xj)'*, 

l = ae-Ud-^ 3c\ 

a b c 

bed 

c d e 

The invariants i, j in the text above differ from these by numerical factors only. 
Thus 

i=:{aby=2I, i = (aJ*)2(6c)2(ca)2 = 6/. 

In connection with the calculation of the values of invariants given symbolically 
in terms of the actual coefficients the reader may find it interesting to discover the 
fallacy in the following : — 

j=(6c)-(ca)2(ai>)2 
= {{bc)(ca)(ab)}^ 
a,^ 61^ Ci2 

0,2' 1)2 c^ 
aj* + bj* + Cj* 

= 27 J. 



ttjOg ijftg *'l<'-> 



206 THE ALGEBRA OF INVARIANTS [CH. X 

Thus the anharmonic ratio is expressed by means of a sextic 
equation in terms of the absolute invariant ^ . 

We see from this equation at a glance that if i = 0, the points 
represented by the quartic form an equianharmonic range, for then, 

p'-p + 1=0. 

Similarly if j = 0, the four points form a harmonic range. 

Again, if two of the points of the range are coincident, one 
value of p is unity : hence 

as it should be. 

172. The anharmonic ratio for the four points 

k/ + \H 

may be obtained at once by writing i^^. for i, and J^^^ for^'. 

To determine those values of « : X, for which the four points 
have any definite anharmonic ratio p; let 

a-21 C^-p-^P^y 

Then i\>.-aj\^ = 0, 

or 3Ha^ + atQ^=0, 

this is a sextic for k : X. 

Now ^n' = -2<n'-n2An. 

Hence {a-6)ta^ = n^Aa, 



«"=±"\/^ 



or .„-.-„, g 

The sextic thus reduces to two cubics. 

If a = 6, it is easy to see that p = 0,l, oo , hence two points 

must coincide. In this case H = 0, and - has one of the three 

A, 

values 

Wi, Wa, m,. 



171-173] GEOMETRICAL INTERPRETATION OF BINARY FORMS 207 

Hence the three members of the pencil for which a = 6 are 

H + m-^f= ^^ 

shewing that if one pair of points coincide, the other pair must 
also coincide. 

If a = 00, the four points form a harmonic range, and ^n = 0. 
There are three members of the pencil for which the range is 
harmonic. If a = 0, then Hq, = ; hence there are only two 
members of the pencil which form equianharmonic ranges. 

In all other cases, there are six members of the pencil having 
a definite anharmonic ratio. 

Ex. If IJx is the Hessian of the cubic a/, prove that the quartic a^Hg. 
is equianharmonic. 

173. Case •when An = 0. The discussion was limited in § 169 
to the case when An is other than zero. Now An is the discrimi- 
nant of the quartic /, and hence when An vanishes, two of the 
roots of /= are the same. We may, as in § 168, write 

where aj' is the square of a linear form ax, and p^^ is a quad- 
ratic. Then as before 

The invariant An is also the discriminant of the cubic 
fl = /r' - ^ «V - ^ V, 

hence, in the present case, fit has a repeated root. Let this be ma, 
and let the other root be nii. Then 

2in2 4- mi = 0, Wg^ + 2mi?7i2 = — « > m^^mi = ~ . 
Hence 

»W2 = - 7 = i (apf, mi = / = - ^ (apY, 



208 THE ALGEBRA OF INVARIANTS [CH. X 

and therefore 

H^m,f=\{pp)Ka^\ 

H + mj= a^' { - i (apYpa,' + 1 (ppf aj'}. 
Again, since 

we obtain as before 

H+mJ=-2<f>% 

where ^ is a quadratic. One of the factors of <f> must be a^., and 
if the other is ^x, then 

The value of t is then seen to be 

t = '^,{ppfax'^x. 
174. We shall now briefly explain another interesting method 
of representing invariant properties of binary forms geometrically. 

If we put Xi = z and x.2 = 1 throughout the work on binary 
quantics the general linear substitution may be written 

az + b . X-, 

z = — -, : , smce z = — . 

cz + a x^ 

Now put z =x + iy, and represent z as the real point x, y 
in the Argand diagram in the usual way ; then the substitution 

az' + h 
cz -\-d 
is a point transformation. 

Unless c = the relation between z and z' may be reduced to 
the form 

{z - a) {z' - a') = k, 

wherein a, a', h are constants. 

Suppose z, z', a, a' are the points P, P', A, A' respectively, 
then the geometrical meaning of the above is 

^P.^'P' = mod. (A;), 

and the sum of the angles that AP and A'P' make with any 
fixed line is constant. Hence the general linear substitution is 
equivalent to an inversion together with a change of origin and a 
reflexion of inclination of the line AP with respect to a fixed line. 
If c = the equation can be written 

/-^^miz-^), 



173-174 a] geometrical interpretation of binary forms 209 

indicating that P' is derived from P by turning BP through 
a fixed angle and increasing it in a given ratio, B being the point 
which represents yS. 

Hence a binary form of order n represents n real points A in 
the plane, and a covariant of the form represents a group of points 
G whose relation to the points A is unaltered by a geometrical 
transformation of the types indicated. 

In particular, the relation of the points C to the points A is 
unaltered by any inversion, because in the particular case in which 
A and A' coincide and k is real, the transformation is equivalent 
to an inversion with respect to A, and a reflexion with respect to 
the real axis through A ; but the properties of the derived 
figure are evidently unaltered by a reflexion alone, and hence they 
are unaltered by an inversion alone. 

174 A. Some of the simpler invariants and covariants can now 
be interpreted. 

If az' + 1hz + c = 0; a'z^ + 2h'z + c' = 

represent the points A, B and G, D respectively, then when 

ac' + a'c - 2bb' = 

A, B, G, D are four harmonic points on a circle. 

In fact on changing the origin to 0, the middle point of AB, 
the first quadratic becomes 

z---k-=- 0, 

and if the second be {z — z^) (z — z^ = 0, 
then since the relation is invariantive we have 

therefore OG.OD^ OA' = 0& 

and OG, OD are equally inclined to OA. 

If we produce GO to D' making OB' = OD, we have 
OG.OD'=OA'=OA.OB; 

therefore GAD'B are coney clic, and by 
symmetry B is on the circle. 

Further as the pencil D' [AGBB] is 
harmonic the four points AB and GD 
form two harmonic pairs on the circle. 
We shall call them harmonically concyclic. 

G. & y. 14 




210 THE ALGEBRA OF INVARIANTS [CH. X 

Ex. Shew that there is one pair of points (P, P') harmonically concyclic 
to each of two given pairs {A, B) and ((7, D). 

Further if {Q, Q') be harmonically concyclic to {A, C), {B, D) and {R, R) 
to {A, D), {B, C), then any two of the three pairs P, P \ Q, Q" ; R, R' are 
harmonically concyclic, 

175. We shall now apply the complex variable to prove 
certain properties of the foci of conies. 

If the tangential equation of a conic be 

AP + ^Hlm + Brri' + 2GI + 2Fm + C=0, 

the axes of coordinates being rectangular, and x^yi be a focus, then 

the line 

a; + yi = a;i + y^t, touches the curve. 

Hence the above equation is satisfied by 1 = , m = , and 

Zi Zi 

we have 

A + -zHl- B -2 {G ■\- Fi) z^ + Cz^^ = 0. 

Consequently the two real foci Zi and z^ are given by the 
quadratic 

Cz'-2(G + Fc)z + {A-B + 2Ht) = 0. 

z +z G F 
Since -^—^ = n + ''n> we see that the centre is the point 
2 

G F 

C C 

Again, if be the origin and ^i, S^ the foci, we have 

SO that the origin can only be a focus when A= B and jET = 0. 

If a system of conies be inscribed in the same quadrilateral, 
their tangential equations are of the form 

and thus the real foci are given by the pencil of quadratics 

All the quadratics are harmonic to the Jacobian (/, /'), and 
accordingly we have the theorem that the real foci of any conic 



174-176] GEOMETRICAL INTERPRETATION OF BINARY FORMS 211 

inscribed in a quadrilateral are harmonically concyclic to a fixed 
pair of points J^, J^. 

We leave the reader to prove that if T^, T^ be harmonically 
concyclic to 8^ and S.2, then the points of contact of the tangents 
drawn from Ti and T^ to any conic whose foci are S^, 82 lie on 
a circle through T^ and T2. Hence if tangents T^P^, T^Q^ be 
drawn to any one of these conies, the circles T^P^Q^ all pass 
through another fixed point. 

Thus the points Ji, J^ are such that if tangents be drawn from 
them to any confocal to a conic inscribed in the quadrilateral, 
then the points of contact lie on a circle through Jj and J^. 

176. The binary cubic. 

Suppose the form is 

az^ + ^hz' + ^cz -\-d = 0, 

and that the points A, B, G representing it are Zi, z^, z^. 

The cubic covariant ^ represents three points A\ B', C on the 
circle ABC such that A, A' are harmonically conjugate to B, G 
and so on. For these three points must be represented by a cubic 
covariant which is not /and therefore must be <}>. 

We have next to interpret the Hessian. 

Let Hi, Hnhe the points representing h^ andAj the roots of the 
Hessian, then we know that for real variables the range {ABGHi] 
is equianharmonic, 

(zi - Z2) (zs - hi) _ (^2 - -gs) (zi - hi) _ (zs - Z i) (Z2 - hi) 
I.e. :j — — . 

Hence since mod (w) = 1, we have on equating moduli 

AB . GHi = BG. AHi = GA . BHi , 

therefore the points Hi , H.^ are the points whose distances from the 
vertices are inversely proportional to the opposite sides. 

To construct them we draw a circle having BG for inverse 
points, and passing through A, with analogous circles for GA and 
AB; then these three circles meet in the points Hi and i/g- 

14—2 



212 THE ALGEBRA OF INVARIANTS [CH. X 

It follows in addition that H^ and H^ are inverse points with 
respect to the circle ABC. 

It is interesting to notice that the Hessian points of A'B'C 
are the same as those of ABC, a fact which gives rise to a curious 
geometrical theorem. 

Another easily proved property of the Hessian points is that 
if H^L, HiM, HiN be drawn perpendicular to the sides, then the 
triangle LMN'va equilateral. 



CHAPTER XI. 

APOLARITY AND ELEMENTARY GEOMETRY 
OF RATIONAL CURVES. 

177. Two binary forms of the same order are said to be 
apolar when the joint invariant which is linear in the coefficients 
of both is zero. 

Suppose that the two forms in question are 

j> = boX-J" + nb^x^'^-^x^ + . . . + hnXa"" = bx"", 

then the only lineo-linear invariant is 

(aby = Uobn - naj)n-i +...+(- l)"an^o 

and this vanishes when the forms are apolar. An immediate con- 
sequence is that a form of odd order is always apolar to itself. 

Thus the discussion of apolar forms may be regarded as the 
development of the theory of the simplest type of invariant ; the 
fact that each set of coefficients only occur to the first degree in 
the invariant renders such a discussion simple and accounts for 
the relative importance of the allied geometrical properties. If 
two quadratic forms are apolar they are harmonic so that we 
may regard apolarity as being, in a certain way, the generalisation 
of harmonic properties. 

The condition for apolarity may be written in other useful 
forms. 

In fact, if the linear factors of <f> are /3x^^\ ^x^^\ ••• /3a;''*', we have 

= [ax\ ^x''' ^*<'' . . . y8^<">}" = (a/S"') (a/3<^0 . . . (a/3<»>). 



214 THE ALGEBRA OF INVARIANTS [CH. XI 

Again, if <f) vanishes for the values 

a?i = S/i""', a^a = ya*""*, r = 1, 2, ... w, 
we have as far as ratios are concerned 

and hence the condition is 

This form of the condition at once shews the connection with polar 
forms ; given the form/, m — 1 of the vanishing points of <^ can be 
chosen arbitrarily and the remaining one is given by polarizing/ 
with respect to each of the n — 1 given values successively and 
equating the final result to zero. 

We can at once find all perfect nth. powers apolarto/, for if all 
the y's are the same we have 

so that y must be a vanishing point of/ Hence / is apolar to 
the nth power of any one of its linear factors, and these determine 
the only 7ith powers apolar to/ 

If/ be apolar to each of the forms ^i, 02> ••• ^m it is apolar to 
^1^1 + ^^2 + . . . + ^jc^k, where the Vs are any constants ; for 

{/>^<^i + ^A+... + Xft<^;fc^=X,(/0i)'^+X2(/<^2r + ...+>^)t(/0A)"=O, 

which establishes the result. 

This result also follows at once from the fact that the equations 
of condition are all linear. 

Again if/ be apolar to each of the (n + 1) forms 

then on elimination of the a's from the equations which they 
satisfy, we find that the determinant of the coefficients of the ^'s 
is zero ; but this is precisely the condition that there should be 
an identical relation of the form 

Xi<^i4- X2^2+ ... + X„+i^,i+i = (A), 

and hence being given n linearly independent forms apolar to / 
any other apolar form is a linear combination of these. It is easy 
to prove directly that n linearly independent forms can be found 
which are apolar to a given form of order n, and this fact will 



177-178] 



APOLARITY 



215 



appear independently from the special system of apolar forms to 
be constructed in the next article. 

178. To determine n linearly independent forms apolar to a 
given one. 

I. Suppose that the factors of the given form / are all differ- 
ent and that except for numerical multiples they are 

ai'^-'o?! + a2"*>a;2, r = l,2,...n, 

then the nth. power of each factor is an apolar form, and they are 
linearly independent. 

For if there be a relation of the type 

1 
where A,,, is a constant, then the (n + 1) determinants of the array 

I ai<''>'\ ai^''^''-W\ «! '''''^-'Oa <»•>,... a2<''>" | r = 1, 2, ... n, 
vanish identically. 

Hence the determinant 



j^^(i)n aj(i)«-Ja2<i), aiW"-2o2<>>S ...tta 



(i)n 



{n)n „ (n)n—i„ (n) ~ {n)n—2„ (?i)2 



(n)n 



Pi\ Pi''^, Px^'^pi, '--P^ 

vanishes for all values of pi and p,^. 

But this determinant being homogeneous in each set of sym- 
bols is equal to 

T—n 

± n (ai<'->a2<*' ~ ai^^'ao**-') n (ai<>2 - a2"''/)i), 

r=l 

where in the first product r and s have all the values 1, 2, ...n, 
but are different. 

Thus if we choose pi,p2 such that -4= —j^v , r=l, 2, ... n, the 

P^ ^ 

determinant can only vanish when for some pair of values of r and s 

ai"*'a2 "'=02 "*'«!'**, 
in which case the two factors d^^'^'^x-^-^- ol.}'^''x^ and ai'*>a7i + a2'*'ir2 
only differ by a numerical multiplier, contrary to the hypothesis 
that the factors are all different. 



216 THE ALGEBRA OF INVARIANTS [CH. XI 

Consequently the n forms 

(«!<'•' a;i + 02 ''•'a;^)", r = l,2,...n, 
are linearly independent when the factors of/ are all different. 

We are thus led at once to the interesting result that the 
necessary and sufficient condition that a form ^ can be represented 
as a linear combination of the nth powers of the factors of y is that 
the two forms should be apolar — the condition is necessary, be- 
cause as each nth power is apolar toy a linear combination of them 
is also, and it is sufficient, because the wth powers are linearly 
independent in virtue of the foregoing, supposing always that / 
has no multiple factors. 

Hence reciprocally, if <f> have no multiple factors, /can be ex- 
pressed as a linear combination of the nth powers of the factors 
of <f). 

This may be regarded as the extension of the elementary 
theorem that all quadratics harmonic to aoc^ + 2hx + c are of the 
form \{x — af + fi{x — yS)^ where a, y8 are the roots of 

ax^ + Ihx + c = 0. 

II. We have still to construct the apolar system when / has 
multiple factors. 

Suppose that /has the factor {ot^Xy^ + a^^'^ replacing r different 
linear factors, and that <^ is apolar to /. Since the relation is 
invariantive, we may change the variables so that the multiple 
factor is simply x/, and then 

«o> c^i> cb^y" dr-i all vanish. 

Hence recalling the condition 

ttobn - na^bn-i + ...+( — l)"an&o = 0, 

we see that it is satisfied for any values of 

^n > Oji_i , . . . On— r+i f 

provided that all the other 6's vanish. 

Thus/ is now apolar to any form for which 

ho,bi,b2, ...bn-r 



178] APOLARITY 217 

are all zero, that is to any form which contains the factor x^^~^^'^. 
Among such we have the r forms 

~ n— r+i/y. J — 1 ^ n—r+ir- »"— 2/y. /y. n—r-v\rr r—\ 

tlUn cCl t M/o *^\ 2 9 * * * 2 2 i 

which are obviously linearly independent. 

Hence, in general, when a form has a linear factor of multi- 
plicity r it is apolar to any form containing that factor (n — r + 1) 
times, and among these apolar forms it is possible to choose r 
which are linearly independent. If each multiple factor be 
treated in the same way we obtain in all n apolar forms, viz. r 
from each factor of multiplicity r; those derived from the same 
factor are linearly independent ; it remains to shew that all the n 
forms are so independent. 

Thus for the sextic x^^ {x-^ + x^- x^ we have six apolar forms in 
three sets 

Xi, x^x^, x^x^ containing the factor x-^*, 

(a?! + x^^x^, {xi + x^^x^ containing the factor {x^ + x^^, 

and x^ derived from the single factor x^. 

The forms in each set are linearly independent, but it has not 
been shewn that the different sets are independent inter se. 

The general proof that the sets so derived are independent 
presents no difficulty but is rather tedious owing to the complicated 
notation. (See Appendix II.) 

Let /= A {xi + ttiiCa)*'' (a-'i + aaa^j)'*^ . . . (ajj + cLj^^'^'p , 

the a's being all different, r^ + r^-^- ... + rp = n, and A a constant. 
This assumes that in / the coefficient of x-^^ is not zero, if it be 
zero we can transform /by a linear substitution into one in which 
x^ actually occurs. 

A form apolar toyis 

and ^-^ -^' ^—^-^ 

are all apolar forms since each involves the factor {x^ + aia;2)"~'*'''"^ 
We shall prove that this set and the corresponding ones derived 
from the other factors are linearly independent. 



218 



THE ALGEBRA OF INVARIANTS 



[CH. XI 



In fact if they are not independent the determinant 
/o(ai), /i(ai), ••• /n(«i) 



foiq), fiiql '■■fniq) 

where in general /« (a^) = a/ 

and /«'(«'-)=a^' 

must vanish for all values of q. 

Except for a non-vanishing numerical multiplier the above 
determinant is the limit of 



/o(ai), 




• 


.•/n(«0 

.fn(ai + t) 




/o(ai + ri- 


■It), Ma, + n- 


~it),. 


-■fn{ai + n- 


It) 




Mq), 


-10, /l(ap + »>- 


-It),. 


'■fniOp + f'P 

-fniq) 


-It) 



^ ^ (l + 2 + ... + r, -l) + (l + 2+ ... +r2-l) + ...(l + 2+ ... +rp -1) 

when t=0. 

But the determinant last written is equal to 

± Jl{af, + pt-a^-a't), 

xll{a^ + ^'t-q), 

where in the first product /a = 1, 2, ... p, 

p' = 0,l,2, ...r,-l, 

o-=l,2,...p, 

o-' = 0,l,2,...r<,-l, 

except that one of the inequalities p^a, p =^a-' must be satisfied, 
and in the second product 

«r = 1, 2, .,.p. 



178-179] 



APOLARITY 



219 



A linear factor is a multiple of t when p = a-, hence t occurs as 
a factor of the determinant to a power equal to 



p 



/3=1,2, ...p, 



and this is precisely the power of t in the denominator. 

To find the limit we put t = when p=^ a- and take the multi- 
plier of t in the remaining factors. 

Besides numerical factors which are certainly not zero, the 
limit is the product of a number of factors of the types ttp — a^- and 
Op — q, and in fact it is easily seen to be 

NU{ap- a^)W<r n (ttp - qYp , 

P+ o- P 

where N is an integer. 

Now the quantities ttj, a^, ... a.p are all unequal and we can 
choose q to be different from each of them, hence the determinant 
does not vanish for all'values of q, and consequently the n apolar 
forms are linearly independent. 

Ex. Evaluate the determinant 



a\ 


o-\ 


a\ 


a^ 


a, 


1 


ba\ 


4q3, 


Za\ 


2a, 


1, 





20a3, 


\2a\ 


6a, 


2, 


0, 





/3^ 


^\ 


^^ 


^\ 


i3, 


1 


5^S 


4^3, 


3/32, 


2/3, 


1, 





f. 


y\ 


■f, 


r^ 


y» 


1 



179. Forms apolar to two or more given forms. 

Suppose we have s linearly independent forms 

fr = ao<'''^i" + na^ ^^^x^''-^x.^ + . . . + ««<*■* a?/*, r = 1, 2, . . . s, 

then the determinants of the array formed by the coefficients cannot 
all vanish, because in that case values of \ which satisfy s — 1 of 
the (n + 1) equations 

\ap <!' + \.^ ap^-^ + ...+ \sap '«' = 

would satisfy all these equations, and hence 

which is contrary to hypothesis. 



220 THE ALGEBRA OF INVARIANTS [CH. XI 

If &o^i"' + nbia;j'^~^X2 + . . . + &n^2" 

be a form apolar to each of the fs, then 

r=l,2,...s. 

These s equations connecting the b's can be solved for s of the 
6's in terms of the others, because, as we have seen, not all the deter- 
minants of s columns formed by the coefficients are zero. Having 
solved the equations we obtain s of the b's expressed as linear 
functions of the others, and as the remaining {n — s +1) b's are 
arbitrary, the general form apolar to all the /'s involves (n — s + 1) 
constants linearly, i.e. there are exactly (n — s + 1) linearly in- 
dependent forms apolar to each of s given forms. 

In particular it follows that there is a unique form apolar to 
each of n given linearly independent forms. 

Further if 

be (n — s + 1) linearly independent forms apolar to the /'s, it is 
clear that every <^ is apolar to every/, and that the most general 
form apolar to each of the <f)'s is a linear combination of the fs ; 
thus the relation between the two sets of forms is a reciprocal one. 

Some interesting results follow from this reasoning. For 
example, given three independent cubics, there is one form apolar 
to each of them, and, if its factors be all different, each of the 
given cubics can be expressed as a linear combination of the same 
three cubes, viz. the cubes of the factors of the apolar cubic. A 
like result applies to forms of any order and constitutes the 
generalisation of the problem of expressing two quadratics each 
as linear combinations of the same two squares. 

The form apolar to the three cubics aj, bi, ci, is 

(6c) (ca) (a6) a^ b^ c^, 

for this is not zero if the cubics are linearly independent and it is 
apolar to d^^ if 

(be) (ca) (ab) (ad) (bd) (cd) = 0, 



179-180] 



APOLARITY 



221 



or, as can be readily seen, if 



ao 


«! 


tta 


as 


bo 


h 


h. 


h 


Co 


Ci 


C.2 


C3 


do 


d^ 


d. 


da 



= 0, 



which is certainly true if dx^ is the same as any of the three 
original cubics. 

The reader may verify that the equation of the apolar form 
may be written 



CtQwi t" vZjtA/2j (Xjt/'j *t" Ct2^2> 1*2^1 "i ^3^2 
60^1 + &1^2> ^1^1 + ^2^2, ^2^1 + &3«2 



= 0, 



which can be easily done by expressing this determinant sym- 
bolically. 

The extension of these results to n forms of order n will pre- 
sent no difficulty. 

180. We may illustrate some of the foregoing results and 
anticipate some of the developments to come by reference to the 
geometry of the rational plane cubic curve. 

Let |, 7], ^ be homogeneous coordinates, then for all points on the 
curve they are rational integral functions of one parameter t and 
of the third order. To apply our results more directly we shall 
replace t by two variables which occur homogeneously, so that we 
have 

pr] = h^x-i^ + ^h^x^x^ 4- 2h^^x^ -\- h^i = /2 U 

and to find the point equation of the curve we must eliminate x^ 
and x.^ so as to obtain a result homogeneous in f, t), ^. But the 
properties of the curve are naturally more easily obtained by using 
the parametric expressions. 

We remark in the first place that the cubics yi,y2,/3 must be 
linearly independent, otherwise all such points ^, 7], ^ lie on a 
straight line ; next the points in which the line 

l^ + mrj -t- n^ = 



222 THE ALGEBRA OF INVARIANTS [CH. XI 

meets the curve are given by the cubic 

which deteraaines their parameters ; hence any straight line 
meets the curve in three points and the curve is therefore of the 
third order. 

Now there is a unique cubic cj) apolar to fi,f2,j3, and (f> is apolar 
to any cubic giving the parameters of three collinear points. 
Conversely if a cubic be apolar to <f) the three points whose 
parameters are determined by it are collinear because it is of the 
form If I + m/a + nf. 

Thus the three points whose parameters are (x^, x^), (3/1, 3/2), 
(^i, Z2) are in a straight line if 

181. Points of Inflexion. At a point of inflexion three 
successive points on the curve are in a straight line, and hence 
the parameters of the points of inflexion are determined by the 
perfect cubes apolar to ^, that is by 

<^ = 0. 

Hence there are at most three points of inflexion, and, since a 
cubic is apolar to itself, when there are three they are collinear. 

But there are other singularities for which three consecutive 
points on the curve are collinear, e.g. cusps, and accordingly we 
shall examine the equation </> = a little more closely. 

If Oa;, /8a!. Ya; be the linear factors of ^ we have 

with similar expressions for rj and ^ ; hence on replacing ^, r), ^ by 
suitable linear combinations — which is tantamount to changing the 
triangle of reference — we shall have 

from which it is readily seen that the straight lines ^=0, i; = 0, 
f = are inflexional tangents, and therefore in this case there are 
three distinct points of inflexion. 

If ^i = has a double factor ^^ we have 

I = \a^' +/i/3a.' + v^ix^ etc., 



180-182] APOLARITY 223 

and hence by a similar transformation we can reduce ^, r), ^ to the 
forms 

I = aa?, V = ^x, f = PxW 

In the neighbourhood of the point 77 = 0, ^ = 0, whose parameter 
is given by ^^ = 0, we see that rj- x ^, and hence this point is a 
cusp, while the line ^ = is an inflexional tangent. The case in 
which ^ = has a treble factor may be rejected because under 
these conditions fi,f2,fz have a common factor and, as will be 
readily seen, the curve breaks up into a straight line and a conic. 

We shall confine the further discussion to the case in which 
the factors of <^ are distinct. 

182. Double point. To each value of the ratio x^ : x^ corre- 
sponds a point on the curve. The same ratio cannot give rise to 
two different points, but the same point may be obtained from two 
different values of the ratio and then it will be a double point on 
the curve, because every straight line through it meets the curve 
in only one other point. We might find the double points directly 
by developing this idea, but the search is best conducted in a dif- 
ferent manner. 

If X, y, z be the parameters of three coUinear points we have 

</>« </>2/ 0z = 0, 

and in general this determines z uniquely when x and y are given. 
When X and y give rise to the same point, and only then, the 
above condition does not determine z but is satisfied for all values 
bi z. 

Now the equation ^j, (py (})z = 

indicates that the quadratic whose vanishing points are x, y is 
apolar to 

the first polar of z with respect to ^a;^. 

Hence if x, y be the parameters of a double point the quadratic 
giving them must be apolar to all first polars of (f)^^. 

Two such polars are 

<f>x'<l>a , ^'x^fi 



224 THE ALGEBRA OF INVARIANTS [CH. XI 

and the quadratic apolar to each of these is their Jacobian, 
i.e. {<t>^') ^x (fi'x <j>a<f>'fi = — {i><f>') <}>x <f>'x ^'a 4>fi, 

<j> and <f>' being equivalent symbols. 
This is 

1 iU') 4>X<i>'x {<f>.<l>', - <^'a0^) = i (HI <f>x^'x iH') («/^) 

and the quadratic required is therefore 

{^^'f 4>x<f>'x = 0, 
namely the Hessian of <f>. 

Thus a rational cubic curve has always one double point, the 
parameters of which are given by the Hessian of the cubic giving 
the parameters of the points of inflexion. 

It will be noticed how readily properties of points on the curve 
are expressed by means of the form <^, and this is natural since ^ 
being given we can write down three forms apolar to it forfi,/^,/^ 
and thence find the ordinary equation of the curve. 

As a further example, we remark that the points of contact of 
the tangents drawn from the point z on the curve to the curve are 
given by 

<f>x'<f>z = 0, 

that is to say there are two such tangents, and the parameters of 
the points of contact are given by the first polar of z. Hence the 
quadratic giving them is apolar to the quadratic giving the para- 
meters of the double points. 

Ex. (i). Prove that the cross ratio of the pencil formed by joining th« 
double point to four points on the curve is equal to the cross ratio of the 
parameters of those four points. 

Ex. (ii). If the parameters of the points of inflexion be given by a^ = 0, 
fix = 0> yx = 0> the point of contact of the remaining tangent to the curve 
from the first is given by 

hence if/ = give the points of inflexion, the cubic covariant gives the para- 
meters of the three points L, M, N, here indicated. 

Ex. (iii). Prove that the six points in which any conic meets the cubic 
are given by a sextic apolar to a given sextic >//■. {i\f is apolar to the squares 
and products /jj/jj/j.) Thence shew that ■\\r = gives the parameters of the 
points where a conic can be drawn having six-point contact with the curve, 
and that inasmuch as these points are the points of inflexion and Z, J/, iT, 
■<\r is the product of/ and its cubic covariant. 



182-183] APOLARITY 225 

183. Apolarity of forms of different orders. In the foregoing 
discussion we have seen that if x and y be the vanishing points of 
the Hessian of a cubic y, then 

a^ayttz = 

for all values of z, so that the Hessian although only a quadratic 
satisfies, in a manner, the condition of apolarity to the cubic. 

If ax, ^x be the factors of the Hessian, we have 

{aa) (a/3) ax=0, 

that is its second transvectant with / vanishes identically (cf. § 91). 

Generalising this we shall call two forms 

f=a^, <^=h^, m>n 

apolar when the nth transvectant 

(ab^ax'^-'' 
vanishes identically. 

Two important facts follow at once from this definition. 

(i) The form f is apolar to any form of order n' ^ m, having 
<ji for factor. 

For if the new form be (f>-\l/ we have 

(f<f>i^r=i(f<f>rH'-^> 

since yfr is of order n — n and the right-hand side vanishes by 
hypothesis. 

A special result is that/ is apolar to any m-ic containing the 
factor ^. 

(ii) is apolar to any polar form of f whose order is n. 

For let tta;" ay"^-^ 

be the apolar form, then 

[ax'^ay'^-'', hx^'Y = {(^^Y ttj/"*"", 

and this is zero since {ahY a^~~^ 

vanishes identically. 

Cor. <^ is apolar to any polar form of / whose order is ^ n 
but ^ m. 

The proof is as above. 

G. & Y. 15 



226 THE ALGEBRA OF INVARIANTS [CH. XI 

184. The conditions of apolarity of 

/= (tto, a, , . . . a^J^x^ , xj^ = a^ 

and (f> = {bo,b^, . . . 6„]i ar^ , a;^)" =6^;" 

are equivalent to m — n+1 linear homogeneous relations among 
the a's. 

In fact, equating to zero the several coefficients in 

(aft)" aa,"*-" 

we have the following equations : 

{aby a,'"-'* = 0, {aby a^"^-^a^ =0, ... (abf ag"^" = 0. 

On being expressed in terms of the actual coefficients the first 
of these relations involves 

do, Oj, ... ttn, 

the second (hfa^, ... a„+i, 

and so on, the last containing 

din—n , Clm—^n+\ > • • • ^n > 

hence if, as without loss of generality we may do, we assume that 
none of the coefficients b are zero, these 

m — n + 1 

relations among the as are obviously linearly independent. 

It follows that by means of them we can express 

{m — n+l) 

of the coefficients a linearly in terms of the remaining coefficients 
and thence that there are 

(m -f- 1) — (m — M + 1) = n 

linearly independent forms of order m apolar to any given form of 
order n less than m. 

185. Construction of a linearly independent set of forms of 
order m(> n) apolar to a given form ^ of order n. 

1. Let the factors of the given form be 

/Si"-'aJi + /S.Mar, = ^^c-), r = 1, 2, ... 71, 

and all different. « 



184-185] APOLARITY 227 

Then the typical form ^^^ is apolar to h^^, for 

and since (/86)" = 0, 

when yS-B is a factor of h^, the result follows at once. 

Next the system of forms 

ye,'*-)", r = l,2, ...n 

is linearly independent, for if there were a relation of the type 

on polarizing (m — n) times with respect to y, we should have 

which is contrary to the established fact that the system of forms 

^^M", r = l, 2, ... n 
is linearly independent. 

II. Suppose next that the factors are not all different but 
that 

Then since the factor /9a; '^' 

for example occurs /Xi times in ^ we know that ^ is apolar to the 
w-ic 

where G is any form of order /u.i — 1. 

In like manner ^ is apolar to the m-ic 

r being any form of order yUj — 1, for 

and this latter is an aggregate of forms each involving the factor 

But since the factor /3a; '^* occurs fi^ times in j> any form involv- 
ing the factor 

vanishes identically. 

15—2 



228 THE ALGEBRA OF INVARIANTS [CH. XI 

Hence choosing any s forms 

1 1, 1 2, ••• 1 s 

of orders /^— 1, /^ — 1. ••• /*«— 1 

respectively, forms /Sa-"* '^' Ft 

are all apolar to ^. 

Next there cannot be a linear relation between them because if 
there were, on polarizing it m — n times with respect to y we 
should obtain a linear relation of the type 

Tt' being of order /i( — 1. This is contrary to what was proved in 
constructing the apolar set of order n. 

Now the form ^^^'^'^''^'^'Ft 

contains fMt arbitrary constants and so we have a form apolar to (}> 
involving 

/*! + /^2 +•••+/*« = W 

arbitrary constants. 

The coefficients of the various constants are each apolar to <^ 
and they are n in number. 

The discovery of forms of order n apolar to a form of order 
m (> n) is a problem quite distinct from the foregoing. 

Suppose / is the given form of order m, then a form of order n 
which is apolar to 

for all values of y will be apolar to f. 

This condition has been shewn to be necessary and it is clearly 
sufficient because if 

then (aby ay^-'' 

vanishes for all values of y. 

Hence the form (f) sought is apolar to the m — n + l forms of 
order n 

and the problem is reduced to one in apolar forms of the same 
order. 



185-186] 



APOLARITY 



229 



If the forms just written down be linearly independent there 



are 



linearly independent forms apolar to each of them, and thus there 
are at least {2n — m) linearly independent forms of order n 
apolar to a given form of order m. There may be more owing to 
the subsidiary system of forms not being linearly independent and 
we shall discuss this question more fully in the sequel. 

It is clear that if ??- > ^ there is at least one form of order n 

apolar to /. 

186. The latter theory finds its natural illustration in the 
problem of representing one or more given forms of order n as the 
sum of a number of perfect nth powers. We shall discuss the 
case of a single quintic at length as an example. 

The general cases present no difficulties — they arise for forms 
of special character. 

Suppose the quintic is 

/= ttoCCi^ + oa^Xi*x^ + lOa^x^xi + lOa^x^x^ + oatXiX^* + (hx^ 

— f, 5_ A 5_. 

The second polars are linear combinations of 

a^ ay ay 

"bx^ ' dxidx^ ' dx^ ' 

If these are linearly independent there is a unique cubic 
apolar to them, and being apolar to all second polars it is apolar 
to the quintic itself. 

On referring to § 179 we may write this cubic in the form 

ay _ay_ _ay^ 

dx-^dx^ ' dx-^dx^ ' ■ dx^dx^ 

_ay_ _ay^ ay 

dx-^dx^ ' dxj)x^ ' dx^ 
or {hcf {caf {ahf a^ h Cx , 

so that it is the co variant denoted by J. 



230 THE ALGEBRA OF INVARIANTS [CH. XI 

Now suppose 

then /= Xa/ + fi^J' + vy^\ 

<^x^xlx being all diflFerent and \, /i, v numerical. 
If^'= a^^^ we have for an apolar quintic 

/= XOa;" + /i«a;*^i + v^^, or a^;^ {'px-^ + (/a^a) + r^^. 
If J = OL^ we have 

/= Xtta," + Ataa;*a7i + x'tta,^ i»l^ or tta,^ (j^aji^' + 25'a;ia72 + rx^). 
This exhausts the cases in which J is not identically zero. 

9Y 02/ 92^ 
If J be identically zero the forms ^^, ^ , —^^ are not 

linearly independent, because if they were they would determine a 
unique non-zero apolar form. 

Lt ^ 5!/" r^=0 

dac^ dx^dx^ dx^ 
be the relation. 

Then we have 

3^" d'f d'f 

" dx^ ^dx^dx^ dx-fixi . ' 

and ay ay ?/!=o 

■'^ dx^dx^ dxidx^ dx} 
and all third polars can be expressed as linear combinations of 

dxj^dx^' dx-fix^' 

If these are linearly independent they determine a unique 
quadratic apolar to both and being apolar to all third polars it is 
apolar to the quintic. 

Suppose this quadratic to be 

(i) aA, then /= Xa/ + /iySa;' ; 
(ii) a^\ then J = a^* {px^ + qx^) ] 
(iii) identically zero, then 

ay ay 

dxi^dx2 ' dxidx^ 
must be identical, hence in this case all third polars are identical. 



186-187] APOLARITY 231 

Now all fourth polars are linear combinations of 

»*/- ^^ (A). 



dx^dx^ ' dx-^dx^ 

93/" 93/" 

But since . xL and - — ~- are identical all the fourth polars 

0x^0X2 dxidx.2^ '■ 

are identical with either of the forms (A). 

Hence the fourth polar is apolar to the quintic, for being of 
odd order it is apolar to itself. 

In this case the quintic is a perfect fifth power unless the 
linear apolar form is zero identically, in which case the quintic is 
also identically zero. 

This completes the discussion and leaves us with six canonical 
forms for a quintic, viz. 

(i) /= \a^' + fi^^' + vy^', 

(ii) y = Xct^' + fi^a^'x, + v^^\ 

(iii) f=\(x^^ + fxa^*x^-\-va^^x^^, 

(iv) / = \a^' + fi^^\ 

(v) /= Xa^« + fia^^Xi, 

(vi) /=\a^^ 

The discussion for any other single form can be conducted in 
an exactly similar manner. 

187. The reader will have no difficulty in applying the method 
explained for the quintic to any binary form ; in particular it will 
be easily seen that, whereas a form of odd order (2w + 1) always has 
at least one apolar form of order (n + l), a form of even order 2n 
has not an apolar form of order n unless the determinant formed 
by the coefficients of its ?ith derivatives with respect to x^ and x^ 
be zero. 

Thus a form of order (2n+ 1) can in general be expressed in 
one way as the sum of (n + 1) (2n + l)th powers, but a form of 
order 2n cannot be expressed as the sum of n 2wth powers unless a 
certain function of the coefficients — manifestly an invariant — be 
zero. For example, in the sextic 

/= ttoXi^ + ea^Xi'^x^ + . . . + tteXo^ = 0/ = 6/ = c/ = dx\ 



232 THE ALGEBRA OF INVARIANTS 

there is an apolar cubic only when 

ao ttj Oa Os 

tti ttj ttj a4 

a, da Cbi 0,6 



[CH. XI 



cu 



a. a. 



a« 



= 0. 



The expression of this as a symbolical form is an instructive 
exercise. There must be a linear relation between 



ay 



s-y d'f 



i.e. between 



ay 



and hence referring to § 179 we must have 

/ = (be) (ca) (ab) (ad) (bd) (cd) ay^^b^Cidd^' = 0. 
Interchanging the letters in ever}' 'possible way we find that 
I = ^^^ (be) (ca) (ab) (ad) (bd) (cd) \ a^, a^a^, a^ai, a^ 

/>8 /)2/t /i/»2 /% 9 

Vl y t/i C/2 J t/i O2 , .1/2 

^1 > ^1 ^2 J vl'iCl'2 J 1*2 

And hence the condition is 

(bey (cay (aby (ady (bdy (cdy = 0. 

The invariant / is called the catalecticant and it will be easily 
seen that a similar symbolical expression holds for the catalecticant 
of any form of even order. 

188. It has been shewn that when 

j = 

identically the quintic can in general be expressed as the sum of 
two fifth powers, and in the course of the work we found the 
conditions under which it can be expressed as the sum of a smaller 
number of fifth powers. A similar process would of course apply 
to any form, but we shall now give a direct answer to the question 
as to what is the smallest number of nth powers in terms of which 
a given binary n-ic can be expressed*. 



See Gundelfinger, Crelle, Bd. c. 413—424. 



187-189] 



APOLARITY 



233 



189. If a binary w-ic can be expressed as the sum of r nth. 
powers it must have an apolar r-ic whose factors are all different, 
so in the first place we proceed to find the necessary and sufficient 
conditions that the form should possess an apolar r-ic. 

If r > — there is always at least one apolar r-ic, 

71/ 

Suppose, then, that r :[> ^ and that there is an apolar r-ic, 

namely <f>. Then <f) is apolar to all derivatives of / whose order is 
equal to or greater than r, and since there are (r + 1) derivatives 
of order n — r, viz. 

ay ay ay 

these cannot be linearly independent. 

Hence there is an identical relation of the form 



ay 



ay 



ay^ 

dxo 



on differentiating this r times with respect to Xi and x^ in the 
(r -f- 1) different ways possible and eliminating the A-'s, we have 

a^y ay a-y 

dx^' dx^^-^dxr, ' '"' a^/a^*- 

a^/ d^f d^f 



dxi^-'dx., ' dxi^-^dx^ 



dx{~'^dx2 



a-/ 



a-y 



av 



= 0, 



aa;/a^/' dx^^-^dx.{+^' '"' 

or say Gr = 0, 

and it is easy to see that Gr is a co variant of y. 

Conversely when Gr = 

by the well-known theorem of Wronski* there is a linear relation 
between 

ay ay ay 

dx/' dx/-'dx^' '" dx/' 

By differentiating this we obtain two independent relations 
between the (r + l)th derivatives, three between the (r-|-2)th 

* See Appendix II. 



234 THE ALGEBRA OF INVARIANTS [CH. XI 

derivatives and in general (p + 1) between the (r + p)th de- 
rivatives. 

Now there are (r + p + 1) derivatives of the (r +^)th class and 
hence of these only r are linearly independent. 

Hence in particular there are only r linearly independent 
(n — r)th derivatives and as these are of order r there is one 
form of order r apolar to them and therefore apolar to the form f. 

Hence when Or=0 there is an apolar form of order r. 

Thus forming the successive covariants 

Go, Gi, G2, ..., 

the necessary and sufficient condition for an apolar r-ic is Gr = 0. 

If Gr-i 4= there is no apolar form of order less than r, for if 
there were any such apolar forms there would be at least one of 
order (r — 1) and Gr-i would vanish. 

Hence if Gr be the first of the covariants G which vanishes the 
lowest order of an apolar form is r. 

Finally if Gr-i 4= 0, Gr = there is only one apolar form of 
order r. 

Suppose in fact there are two apolar forms of order r and that 
for simplicity their factors are all different in both cases. 

Let (a.1 + agiPg), s=l,2,, . . ., r 

be the factors of the first, and 

(xi + ^.x^), 5=1, 2, ..., r 

be the factors of the second, then we have 

r r 

1 1 

therefore there is an identical relation of the type 

r r 

1 1 

Now 2r is less than w, hence by § 178 such a relation is only 
possible when the coefficients of the various 7ith powers severally 



189-190] APOLARITY 235 

vanish ; thus since the a's are all different and the yS's are all 
different it follows that either every X and every jjl is zero or else 
for a certain number t of values of s 

while for other values of s 

Consequently f can be expressed as the sum of t nth powers 
where t<r, hence there is an apolar form of order t. But in this 
case we must have Gr-i = contrary to hypothesis, hence there is 
only one apolar form of order r. 

The reader will easily establish the fact that if there are only 
two apolar r-ics they must have (r— 1) common factors and 
that these factors multiplied together give the apolar (r — l)-ic. 
Further the extension of the above to the case in which two or 
more a's are equal will present no difficulty. 

Cor. Since Gi is the Hessian of/ the necessary and sufficient 
condition that / should be a perfect nth power is that its Hessian 
should vanish identically. 

Ex. (i). If / is apolar to ^ then / is apolar to every form having ^ 
for factor. 

Ex. (ii). If a binary form of order n have an apolar r-ic {r < %), then it 
has at least (s + 1 ) independent apolar forms of order r + s. 

Ex. (iii). Shew that the argument of § 188 can be extended to any number 
of binary forms, and construct a table of canonical forms of a simultaneous 
system consisting of a cubic and a quartic. 

Ex. (iv). Shew that in general two forms of orders n^ and n^ can be 
expressed as linear combinations of powers of p linear forms if 

3jo — 2 = Wi + »2 ; 

find the jo-ic giving these linear forms and extend the results to any number 
of forms. 

Ex. (v). From the symbolical form of a catalecticant deduce the sym- 
bolical forms of the covariants G. 

190. We shall conclude this chapter with a few geometrical 
illustrations of the foregoing theory. 



236 THE ALGEBRA OF INVARIANTS [CH. XI 

Binary Quadratics in connection with the Geometry of a Conic. 

If we have the equations 

^ = a^Xi^ + la^XyX^ + a^x^ = a^ =/, 

then the point |, tj, ^ lies on a fixed conic. 

The equation of the conic is easy to find. For the line 

touches the curve when 

Disct. {Xf+ fi<f> + mjr) = 0, 
i.e. if Vt'ii + /i^*2-2 + ^*33 + 2/iM23 + ^vXigi + 2XjjLii2 = 0, 
where t'la = (/, <f>y etc. 

This being the tangential equation the point equation is 

*11 *12 *13 s I ~ "> 

1l2 ^22 *2S V 
*13 ''^23 *33 b 

I ^7 r , 
or 7„ f= + 72217^ + 733 r^, + 21^7}^ + 2T^ ^1 + ^J'u^ = 0. 

where /,« is the minor of v« in the determinant 

hi *12 ^3 
*ia *22 *23 
*1S *2S *33 

In particular the equation of the conic gives an identical 
relation between three quadratic forms and their invariants. 

191. An immediate inference from the parametric expressions 
for ^, 7), ^ is that the cross-ratio of the pencil joining a variable 
point P on the conic to four fixed points x, y, z, o> on the curve is 
equal to the cross-ratio of the parameters of those four points, i.e. 

(xy) (zo)) 
{xo) (zy) ' 

For let the equations of two lines through P be X = 0, Y=0 
and let X+tY=0 be the equation of the line joining P to the 



i 



190-192] APOLARITY 237 

point X on the curve. Then there is an algebraic relation con- 
necting t with -^ and since to one value of i corresponds one value 

of — and vice versa we must have 
x^ 

_ Ax-i + Bx2 

~ Cxi + Dx2 ' 
where A, B, C, D are constants. 

Hence the cross-ratio of four values of t is equal to the cross- 
ratio of the corresponding four values of the parameter — . 

Xt^ 

192. In connection with this conic there is a simple corre- 
spondence between binary quadratics and straight lines in the 
plane, for a binary quadratic % equated to zero gives two points 
P, Q on the conic, so that if we make % correspond to the line PQ 
when either is given the other is uniquely determined. 

The quadratic % can be written in one way in the form 
\f+fi(}> + vylr, 

and then \^ + fir) + v^=0 is the equation of the corresponding 
straight line. 

Accordingly if the line pass through a fixed point in the plane, 
say |o. Vo, ?o, we have 

= \f+fi<f> r^ ^|r 

bo 



= '^(/- 



l>)+K^-|>) 



and 'x^ is apolar to (i.e. harmonic to) the fixed quadratic apolar to 
/-|Vand(^-|?t. 

bO bo 

Hence if the line PQ passes through a fixed point T, the 
corresponding quadratic '^^ is harmonic to a fixed quadratic t. 

Now the perfect squares apolar to t correspond to lines 
touching the conic, since in this case the points P and Q coincide, 
hence these perfect squares determine the points of contact of the 
tangents drawn from T to the conic. 



238 



THE ALGEBRA OF INVARIANTS 



[CH. XI 



If these points are R, S, then the quadratic giving R and S is 
apolar to the square of the linear forms giving R and »S respec- 
tively, and since J is apolar to these latter squares, it follows that 
J corresponds to RS, as can be seen in many ways. 

Thus if PQ pass through T, and jR^ be the polar of T, the 
quadratics corresponding to PQ, RS are harmonic, or in other 
words, when two quadratics are harmonic, the corresponding lines 
are conjugate with respect to the conic. 

This can be easily verified by using the tangential equation 
of the conic. 

Consider a binary quartic representing four points A, B, C, D 
on the conic, and let BG, AD meet in E, CA, BD in F, and AB, 
CD in Q. 




Then the quadratic corresponding to the polar of E is harmonic 
to the two quadratics corresponding to AD and BG, i.e. this polar 
meets the conic in two points which are the double points of the 
involution having B, G and A, D ior conjugate elements. 

Thus the polars of E, F, G meet the conic in the double points 
of the three involutions determined by the four points A, B,G,D; 
but EFG being a self-conjugate triangle of the conic, the polar 
of E is FG, and the lines EF, FG for example are conjugate lines 
with respect to the triangle, hence the pairs of double points of 
any two of the three involutions are harmonically conjugate. 

[This corresponds to the fact that the sextic covariant of a 
quartic can be written as the product of three quadratics which are 
mutually harmonic] 



192-193] APOLARITY 239 

193. As another example, let us prove that a triangle and its 
polar triangle with respect to the conic are in perspective. 

Suppose the sides of the triangle A1A.2A3 correspond to the 
quadratics /j, /g. fs respectively, and that the sides of the polar 
triangle B^B^B^ correspond to ^i, ^2. ^3 respectively. Then since 
jBjjBj is conjugate to A-iA^ and A^A^, it follows that <^3 is harmonic 
to ^ and f■^, so that </)s is the Jacobian of fx and f^. 

Hence the sides of B^BiB^ correspond to 

Jts, J 31, and Jia where J^^ = {f^f^). 

Now let B^B^ meet A-^A^ in Pj, then the polar of Pj is 
conjugate to both these lines, and therefore corresponds to 

(<^i,/i)orto (/i, J23). 

The polars of the analogous points Pg, Pg correspond to 
(/a, Jzi), (/s, J12) respectively, and Pj, Pg, P3 will be collinear 
if their polars are concurrent, i.e. if the quadratics 

KJly *J^), (/2, ''Sl/j \fz^ J 12) 

are harmonic to the same quadratic. 

To prove that this is so, let us calculate the quadratic harmonic 
to the first two. 

Representing 

J\> J2> y 3 Oy eta; > Oa; , CjB , 

we have J23 = (J>c) h^Cx, etc. 

.'. (/i, Jo^)' = [ax, (be) hcx] = i (ac) {be) aj>^ + ^ (ab) (be) a^c^ 

= i {{acY bx' + {bey ax' - {abf Cx'] - \ {{abf Cx' + {bcf a^^ 

-{acfbx'] 
= \{acfbx'-^{abycx'. 

Similarly 

if. , J31) = i {aby Cx' - i {bey ax', 

and the quadratic harmonic to these two is 

{{acy bx' - {aby Cx\ {aby Cx' - {bey aj^], 
or {aby {acy J^ - {aey {bey J^ + {aby {bey /31 , 

or {aby {acy J^ + {bay {bey J^ + {cay {eby J,„ 

and by symmetry this is harmonic to {f^, J^^ also. 



240 THE ALGEBRA OF INVARIANTS [CH. XI 

Thus the polars of P1P2P3 meet in a point whose polar 
corresponds to 

*23 *31 *12 

where 1*23 = (f^, f^f = (bey. 

And hence P^PJPz lie on the straight line corresponding to 
the quadratic 

1^ I3I t>i2 

The reader will find it interesting to shew that the above 
quadratic may be written 



T T T ~ ' 



where the I's are derived from the is as before. 

Ex. (i). ABC is a triangle inscribed in a conic, and the tangents 
at J, B, Cmeet BC, CA, AB in Z, Jf, N. Determine the parameters of the 
points in which the polar of L meets the conic in terms of those of A, B, C. 
Hence shew that Z, M, N are coUinear, and that the line LMN meets the 
conic in the Hessian points of ABC. 

Ex. (ii). Deduce Pascal's Theorem from the parametric representation 
of a conic. 

Ex. (iii). Prove that six conies can be drawn through four fixed points to 
touch a given conic, and that their points of contact are given by the Jacobian 
of the quartics determined by any two conies through the four points. 

194. Twisted Cubic. The relation of the rational cubic 
space curve to the binary cubic is exactly the same as that of the 
conic to the binary quadratic. 

The line of argument is similar to that used in the previous 
case. 

We have 

1^ = a^x^ + SajiCi^dJa + ^a^XiX^ + a^x.^ = ai =/, 
rj = bj' = <f), 

and the curve represented is a cubic because the plane 

X^ + fjbT] + v^ + prs = 
meets it in three points. 



193-195] APOLARITY 241 

We have now an exact correspondence between planes and 
binary cubics. If a plane pass through a fixed point T the 
corresponding cubic is apolar to a fixed cubic which determines 
P, Q, R the points of contact of the osculating planes of the cubic 
through T. Cf. § 192. 

Hence there are three such points, and since a cubic is apolar 
to itself, the plane joining them passes through T. We shall call 
T the pole of the plane PQR; and we observe that when two 
cubics are apolar, each of the corresponding planes passes through 
the pole of the other. 

If TLM be a chord of the cubic through T, then any plane 
through this line is apolar to the cubic giving PQR, so that L, M 
are determined by a quadratic apolar to the cubic, i.e. by the 
Hessian of the cubic. 

195. We shall illustrate the above remarks by the discussion 
of an algebraical problem intimately connected with the cubic 
curve. 

The planes passing through a fixed line I correspond to cubics 
of the form 

where \ and X^ are variable numbers. We call such a linear 
system of cubics a pencil, and then we see that the Jacobian of 
two members 

of the pencil is 

J = (Xlf^2 — \2fll) (/1/2), 

so that except for a numerical factor it is the same for every pair 
of cubics in the pencil. Hence we may call this quartic the 
Jacobian of the pencil and the question arises — Given a binari/ 
quartic J, can a pencil of cubics be found of which it is the 
Jacobian ? 

To answer this question consider the geometrical meaning of 
the Jacobian with reference to the- line I. 

If the cubic \fi + X.2/2 

have a double factor this factor occurs in the Jacobian, because it 
occurs in both 

G. & Y. 16 



242 THE ALGEBRA OF INVARIANTS [CH. XI 

And further, as the discriminant of a cubic is of degree four 
there are only four members of the pencil which have a double 
factor; therefore the Jacobian of the forms contains these four 
double factors and no others. 

But if a cubic Xj/i + X^/a 

have a double factor the corresponding plane touches the curve, and 
hence through a given line four planes can be drawn touching the 
curve, and their points of contact are given by / = 0; in other 
words there are four tangent lines to the curve which intersect a 
given line I, and they are tangents at the points given by 

J=0. 

Now our problem is equivalent to the following : Being given 
the four points of contact, to construct the line I. But the four 
tangents being given there are two lines meeting them, and, as each 
of these corresponds to a pencil of cubics having / for Jacobian, it 
follows that the question is always soluble and that there are two 
solutions which may coincide in particular cases. 

Ex. (i). Prove that three osculating planes and one chord of a cubic can 
be drawn through any point. The chord determines the Hessian of the cubic 
giving the points of contact of the planes. 

Ex. (ii). The plane joining the points of contact of the three osculating 
planes from a point passes through and is called the pt)lar plane of 0. 
If the polar plane of passes through (Z the polar plane of (7 passes through 
and the corresponding cubics are apolar. 

Ex. (iii). By using line coordinates prove that the tangents to the cubic 
belong to a linear complex and that any two planes cori-esponding to apolar 
cubics meet in a line belonging to this complex. 

Ex. (iv). If a line I belong to the above complex the points of contact of 
the tangents to the cubic which meet I are given by a quartic for which the 
invariant i vanishes. In this case the two pencils of cubics having the 
quartic for Jacobian coincide. 

Ex. (v). If two pencils of cubics have the same Jacobian, then any 
member of the first i)encil is apolar to any member of the second i^eucil. 

196. The twisted quartic. The rational space curve of 
the fourth degree furnishes the most convenient geometrical 
representation of a binary quartic and its concomitants. Suppose 
that 

^ = Px\ V = qx\ K^'Tx, ^ = t,* 



195-197] APOLARITY 243 

is the parametric representation of such a curve. Then since 
^, 7j, f, zT are not connected by a linear relation the four binary 
quartics are linearly independent and are therefore all apolar to a 
unique quarticy which we shall denote by 

Ux — ^x — ^x — ^x • 
It is evident that four points on the curve are coplanar when 
and only when the quartic determining their parameters is apolar 
to f — in fact such quartics are linear combinations of f, i], ^, tsr. 
Hence the four points a, /3, 7, 8 are coplanar when 

aattpaytts = 0. 

Thus ttx* = gives the four points of superosculation, i.e. the 
four points at which the osculating plane contains four consecutive 
points of the curve. 

Through a point 8 on the curve can be drawn three osculating 
planes other than that at B and their points of contact are given by 

cix'^cis = 0, 
i.e. by the first polar of B. 

By varying B we obtain a pencil of cubics each of which 
determines three points on the curve such that the osculating 
planes at them meet in a point on the curve. 

Now if a member of this pencil have a double factor we have 
two consecutive osculating planes whose line of intersection meets 
the curve, and hence the double factor gives a point the tangent 
at Avhich meets the curve again. But the double factors of 
members of the pencil are precisely the factors of the Jacobian 
of the pencil, which in this case is the Hessian of f. Hence 
11 = gives four points on the curve the tangents at which 
meet the curve again. 

197. We have still to interpret the sextic covariant which 
gives six points on the curve. 

For this purpose let us seek for points P, Q on the curve such 
that the osculating plane at P passes through Q and the osculating 
plane at Q passes through P. 

If X, /i, are the parameters of P and Q we have 

a\^a^ = 0, 

16—2 



244 THE ALGEBRA OF INVARIANTS [CH. XI 

To eliminate \ we note that from the second equation 

and as \ occurs to degree three in the first equation we must use 
three different equations of this type. 

Hence we have 

(ah) (ac) (ad) a^h^^c^^d^^ = 

as the equation for fjb. 

Now 

(ab) (ac) (ad) axhiCx^d^^ 

= \ (ab) aj)^c^d^ {(acj d^' + (adf c^ - (cdj a^] 

= \ (ab) (acf a^bic^ •d^-\-\ (ab) (adf axbid^- . c^ 

-\(ab)a^^bi.(cd)''CxH^ 

where t is the sextic covariant. 

Fory=0 the points P and Q coincide, hence there are three 
pairs of points such that each lies in the osculating plane of the 
other and they are given by the sextic covariant. 

198. We shall now sketch some further investigations con- 
nected with the curve. 

The triple secants of the curve are given by the pencil of cubics 
apolar to/, so that if one cubic give three points on a triple secant 
and another three points whose osculating planes meet on the 
curve the two cubics are apolar. The two pencils have the same 
Jacobian, as follows from geometry and analysis. 

The four points in which the tangents at the Hessian points 
meet the curve again are given by 

(ahY ((nc) (bd) (cdy ajy^c^d^ = 0, 

for the discriminant of their first polars must vanish. 

This reduces to iH —jf= 0. 

If t = the four points of superosculation are coplanar. If 
j = there is an actual double point on the curve, because in that 
case / has an apolar quadratic — the quartic is now the complete 
intersection of two quadrics which touch, whereas in general it is 
the partial intersection of a quadric and a cubic. 



197-198] APOLARITY 245 

There are two other tangents meeting any given tangent to 
the curve and when j = there exist pairs of chords such that 
the tangents at the extremities of either intersect those at the 
extremities of the other. 

Between the parameters of the points of contact of two 
intersecting tangents there is a symmetrical (2, 2) relation, hence 
by comparison with Poncelet's porism we infer that if one twisted 
polygon with n sides can be circumscribed to the curve there is 
an infinite number of such circumscribing polygons. 

By forming the expressions for the six coordinates of the 
tangent at any point, which are the Jacobians of the fundamental 
quartics taken in pairs, we see that the condition that six tangents 
should belong to the same linear complex is that the sextic 
determining their point of contact should be apolar to a fixed 
sextic. This fixed sextic can be shewn to be the sextic covariant 
either geometrically or analytically. We can in fact establish the 
following theorem : If two quartics are apolar to a third then their 
Jacobian is apolar to the sextic covariant of the latter. Ex. (ii), 
p. 52. 

The sextic covariant therefore bears the same relation to the 
line geometry of the curve as the fundamental quartic does to the 
point geometry. 

By discussion of the quartics apolar to this sextic the reader 
will easily prove that the tangents at four points given by a 
quartic covariant (of the type Xilf + fijf) are generators of a 
quadric. 

References to various memoirs of Reye, Rosanes and others on the subjects 
discussed in this chapter will be found in Meyer's Berichte. See also the 
same author's book Apolaritat und rationale Curve. There is an interesting 
paper on the twisted cubic by Sturm in Crelle^s Journal, Vol. Lxxxvi., and a 
very full list of references for the twisted quartic in a paper by Richmond, 
Camh. Phil. Trans. 1900. 



CHAPTER XII. 

TERNARY FORMS. 

199. In this chapter we shall extend the symbolical notation, 
which has been used throughout for binary forms, to the case 
of forms with a higher number of variables. It may be well to 
remark at the outset that the methods developed up to the 
present are nowhere else so effective as in the case of binary 
forms; accordingly, while we shall explain at some length the 
notation and methods for ternary forms, we shall content our- 
selves with a very brief indication of the extension to forms with 
four or more variables. 

200. A ternary form in one set of variables x^, x^, x^ will be 
written in the form 

/ = S , .^., apqr x,PxM/ ; p + q + r=n, 

and the summation is of course extended to all possible values 
of the integers p, q, r satisfying this condition. 

In agreement with the symbols previously introduced, we shall 
represent y* by the umbral expression 

so that a-^PaSoL^ — apqr. 

Just as before an expression in the a's has no actual (as 
opposed to symbolical) significance unless the total degree in 
the a's is exactly n. To represent forms whose degree in the a's 
is greater than unity, we have to introduce other equivalent 
systems of symbols /3 and 7, so that 

J — "a; Hx — v* • • • 



199-201] 



TERNARY FORMS 



247 



201. Let us at once point out how concisely polar forms 
are represented in this notation ; the rth polar of yiy^ys with 
respect to / is 

in-r)\{ d d dy. 

the numerical factor being introduced for arithmetical conve- 
nience. 



This expression is 



{n-r)\( d_ d_ 

" dx^^y^bx, ' ^'dx.. 



(n — r) ! n ! 
n ! {n — r) 



y^^+y^uz+y^T^^) («i^^ 



+ OsaJj + agajs)'* 



a "'-''a »■ 



i.e. the rth polar is represented by 



Thus for example the ecjuation 

«/ = 

represents a curve of order n, and if 3/ be a point on it the equation 
of the tangent at that point is 

axOy''-' = 0. 

Again, the points of contact of the tangents drawn from a 
point y to the curve lie on the first polar 

Next let us consider the effect of a linear substitution on a 
form such as f. 

We shall write such a transformation in the form 

^1 = ll^l + »7l^2 + fl^S \ 
«2 = la-^Ti + 7/2X2 + ?2^3 >• 
X3 = ^jZi + V3X.2 + ^3^3 J 

The effect of this change of variables on olx is to change 
it into 

ai(|i^i+^iA^+ ^,X,)+u.,{^,X\ + rj,X,+^,Xs) + a,(^,X,-^v3X,+ ^,X,), 

or a^Xi + a^X., + a^Xj . 



248 THE ALGEBRA OF INVARIANTS [CH. XII 

Hence / which is cn^^ becomes 

or S , ; , a^a'ia/XyPXsX/, where « + g + r = n. 

Thus the coefficient of Zi" is a/*^ or the result of putting 
the l^'s for the ip's in/, and subsequent coefficients may be deduced 
from this by means of the polarizing operators 

operating on the first. 

202. The above will shew how very convenient the symbo- 
lical notation is for dealing with constantly occurring functions 
like polars, but, as in the case of binary forms, its great value lies 
in the application to the theory of invariants and covariants. 

Let us recall the significance of these terms especially for 
ternary forms and illustrate them by some examples. 

Suppose that after the application of a linear transformation 

Xr = Ir^i + rj^X^ + ^r^s. ^ = 1, 2, 3, 

a ternary quantic 



p\q\r 



f- ^^w \^\„\„\ ^^i^^aW 



becomes F = 2^„„^ , ; , XfX^X^, 

^ 'p\q\r\ 

then a rational integral function / of the coefficients is said to be 
an invariant when 

where M depends only on the transformation. 

A covariant of / possesses a similar property, but involves the 
variables as well as the coefficients, and similar definitions apply 
to invariants and covariants of two or more forms. 

After what has been said on the subject in dealing with 
binary forms, we need not stop to prove that it suffices to 
consider invariants and covariants which are homogeneous in 
the coefficients of each form involved, and covariants which are 
homogeneous in the variables — further it may be easily proved 



201-203] 



TERNARY FORMS 



249 



that the factor M referred to in the definition must be a power 
of the determinant of the transformation, and we shall assume 
henceforth that it is so. The reader may regard this as a simplifi- 
cation of the definition, or supply the necessary proof on the lines 
of that given for binary forms. See § 24. 

As examples, we may notice that the discriminant 
ahc + %fgh — af^ — hg- — ch^ 
is an invariant of the ternary quadric 

awi^ + hx2 + cxz + ^fx^s + ^g^^i + 2hxiX.2. 
In this case the power of the determinant which occurs as 
multiplying factor is two. 

Again the Hessian. 

BY 



dxi^ 

dx^dxi 
BY 



ay 

dxidx2 
dx2 

ay 



jy 

dx-^dxs 

ey 

dx^dx^ 

ay 

dx^- 



dx^dxi dxs dx2 

is a covariant of any ternary form /. 

As a further example, the Jacobian of three forms is a covariant 
of the three forms. 

We shall not stop to verify that these expressions actually are 
invariants or covariants as the case may be. They are well known 
in the theory of curves and we mention them here to remind the 
reader that such invariant functions as defined above do actually 
exist. By the use of the symbolical notation we shall be able to 
verify our assertions much more easily, and also to construct any 
number of invariants and covariants. 

203. Just as in the theory of binary forms we denoted the 
determinant 

I «! a., by (a^), 

I A A 

so now we shall denote 

«! Oo a, I by (a^y). 
/3x /3. /33 j 
7i 72 73 i 



250 THE ALGEBRA OF INVARIANTS [CH. XII 

We are now in a position to enunciate and prove the first 
theorem connecting invariants and symbolical notation, namely : 
Every expression represented symbolically by factors of the type 
(a^y) is an invariant, and every expression represented by factors 
of the types (a^y) and a^ is a covaHant. 

For by a linear substitution a^ becomes 

and hence {(x&y) becomes 

i.e. (a/37)(|770. 



Qf 


a, a^ 


A 


^r, ^C 


7^ 


yv 7f 



So that the expression formed from the new coefficients is equal 
to that formed from the old coefficients multiplied by a power 
of (^r/^) ; and the power of (^v^) that occurs is equal to the 
number of symbolical factors of the type (a^^Y) that occur in the 
expression, 

204 The preceding proof, depending only on the way in 
which the symbolical letters occur in the expression under 
consideration, applies equally well to an invariant or covariant 
of any number of ternary forms. The only further condition that 
such a symbolical expression should actually represent a covariant 
is that it should have a meaning when expressed in terms of the 
original coefficients, i.e. each symbolical letter must occur to the 
requisite degree in every term in which it appears. It is easy to 
verify that the three examples already given are actually invariants 
or covariants as the case may be. 

Thus if 

Oaoo^i'^ + ao!!o^2^ + aoo-2^3^ + ^ani/ViXz + 2aioia;3^i + 2aou«'3p^3 

= Ctx- = ^x' = yx, 

we have by direct multiplication 

(a/87)' = (2±oA73)^ = 6 



^200* 


^110 > 


C^lOl 


^110 > 


Cto20> 


ftoJl 


ttioi» 


^011 J 


Ct002 



since a.-^ = fii^ = y^ = a^oo , etc. 

Thus the discriminant of the ternary quadric is an invariant. 



203-205] 






TERNARY FORMS 




Again, if we 


have three forms 




then 




3-1= aia^".-i, a,aa;"i-i, 


tta"^"'-^ 








A/3^'^^-\ /3./8^'^-\ 


Mx''^-' 








7i7/''"'. 727*"'"S 


7a7x"=-^ 


1 


3 


8^1 


9/i 
8a?2 


5/i 


^ A SC/iA/a) 


njU^n 


n^natig 8 {x 


l*'^2^3^ 








a/; 

9a;2 


3^3 












dx.2 








which shews at 


on 


ice that the Jacobia 


n is a CO variant. 



251 



The fact that (ot^yf ax^'-^^x'^'^Jx''^ represents the Hessian of 
the form /= a^'^ = /3a;" = jx^ niay be easily verified in the same way. 
The deduction of the equation of the Hessian from the property 
that the polar conic of any point on it degenerates into two straight 
lines affords an instructive example of the symbolical calculus. 

The polar conic of the point yi, y^, y-i with respect to the 
curve 

/= Oix'' = ^x'' = 7x" = 

is represented by the equation 

and the discriminant of this is 

Hence changing y into x we get the Hessian as the locus 
of points possessing the property in question. 

205. Geometrical Meaning of a Linear Transformation. 

To obtain a geometrical representation of ternary forms we naturally 
suppose the variables x^, x^, x^ to be the homogeneous coordinates 
of a point in a plane referred to a certain triangle. The equation 
to zero of a ternary form will then represent a curve in the plane. 

A linear transformation 

Xr = t,X, + rj,X, + t:,X,, r = 1, 2, 3, 

may be regarded from two points of view, for if P be the point 



252 THE ALGEBRA OF INVARIANTS [CH. XII 

sCi, x.i,Xz we may either suppose P to be unaltered and the triangle 
of reference changed or we may suppose P to be changed and the 
triangle of reference unaltered, in other words X^^X^, X, may be the 
coordinates of the original point referred to a new fundamental 
triangle or they may be the coordinates of a new point (into which 
P is changed) referred to the original triangle of reference. 

The general linear transformation as written above can be, in 
fact, represented as a change in the triangle of reference together 
with a multiplication of each coordinate by a suitable quantity — 
the equations of the sides of the old triangle of reference referred 
to the new triangle are Xr = 0, that is 

^rX^ + 77rXj + frXs = 0, r = 1, 2, 3, 

and on solving for X^, X^, X^ in terms of x^, Xo, x^ we can easily find 
the equations of the sides of the new triangle of reference in the 
old coordinates. 

From this point of view, if I be an invariant of a ternary 
form f, when / vanishes for one triangle of reference it vanishes 
for any other triangle of reference ; that is to say, 7=0 expresses a 
property of the curve itself and not a relation between the curve 
and some particular triangle. 

In like manner the curve represented by a covariant is con- 
nected with the original curve by relations which are quite 
independent of the triangle of reference. 

From the second point of view the point P is changed into a 
point P' by a homographic transformation, i.e. a transformation 
possessing the following properties : 

(i) any point P is changed into a point P ; 

(ii) any straight line 'p is changed into a straight line /;'; 

(iii) if P lies on 'p then P' lies on p ; 

(iv) the cross-ratio of four collinear points is equal to the 
cross-ratio of the four corresponding points ; and the cross-ratio of 
four concurrent straight lines is equal to the cross-ratio of the 
four corresponding lines. 

Of these properties (iv) is the only one we need prove. 

Let jpi = and P2 = be the equations of two straight lines and 
p/ = 0, p2=0 the equations of the corresponding lines, then 



205-206] 



TERNARY FORMS 



253 



Pi +\p.2 =0 corresponds to pi + \p2 = for all values of X, and 
inasmuch as the cross-ratio of the four lines 



Pi + \rP2 = 0, r = l, 2, 3, 4 



IS — 



(X.2 — Xs ) (Xi — X4) 



the result follows at once. 



(X3-X,)(X2-X4) 

Again, suppose that a figure in one plane is projected into 
another plane and that P' is the projection of F, then if x^, x^, x^ 
be the coordinates of P referred to a fixed triangle in the first 
plane and Xj, Xo, X3 be the coordinates of P' referred to a 
triangle in the second plane, it is very easy to see that x^, x^, x^ 
are linear functions of X^, X^, X3 and vice versa. On the con- 
nection between projection and linear transformation see § 157. 

Thus projection affords an example of linear transformation 
and in this case the four properties enunciated above are self- 
evident. 

Hence if / be an invariant of a form /, then 7 = expresses 
a property of the curve /= that is unaltered by any homo- 
graphic transformation and in particular by projection. A like 
remark applies to the connection between a covariant curve and 
the original curve — it is undisturbed by projection. 

206. The connection between these two different points of 
view is interesting. 

The linear transformation 

Xr = ^rXi + rjrXn + ^^Zj 

leaves three points of the plane unaltered in general. In fact to 
find the points whose position is unchanged we have merely to 
put Xr = px,. in the equations above and then elimination of x^, x^, x^ 
1 



gives 



5 a cubic for 



- , VIZ. 

P 



Vi, 



V2- 



Vi> 



^3- 



=0. 



254 THE ALGEBRA OF INVARIANTS [CH. XII 

lu general this equation has three unequal roots and to each 
root corresponds a single determination of the ratios x^ : x^ : x^. 
If we take the triangle formed by these three points for triangle 
of reference the linear transformation takes the simple form 

00^ — - aTj ,A j y U/2 ^~ A^2 "^ 2 ) ^H ^~ "^3 "^ 3 • 

And hence in general by a suitable choice of the triangle of 
reference a linear transformation can be reduced to this form in 
which the coordinates are only multiplied by constants. 

Ex. (i). Prove that by suitably choosing the coordinate system any four 
points i^nay be reduced to the form 

^.^(i) = 0, A-3(i) = 0, .Vj(-^) = 0, A-3(2)=o, :r,(3)=0, a:p=0, 

And hence prove that a linear transformation can be found which changes 
any four points into four given points. 

Ex. (ii). Given four points and their four con-esponding points, shew that 
the point corresponding to any fifth point can be constructed by means of the 
ruler only. (Use the cross-ratio property.) 

Ex. (iii). Hence or otherwise shew that any linear transformation is 
equivalent to a ruler construction. 

Ex. (iv). If the linear transformation leave an isolated point and every 
point on a fixed line unaltered, shew that the equation of § 206 has a double 
root which is a root of every first minor. Give an equivalent geometrical con- 
stmction in this case. 

207. Starting with the definitions of invariants and covariants 
already given we might proceed to prove that every invariant and 
covariant can be represented symbolically in the form indicated 
in a previous theorem, and then go on to develop a theory of 
invariants and covariants as far as possible on the lines of binary 
forms. However it is essential to remember that the real and 
primary importance of such a theory lies in its application to 
geometry, and as in geometry line coordinates and tangential 
equations occur quite at the beginning of the analytical ex- 
position, we are led here to introduce line coordinates as well as 
point coordinates from the first. Indeed it is not difficult to shew 
on purely analytical grounds that line coordinates are essential for 
the proper treatment of the algebraical questions that arise ; such 
an explanation together with a comparison with the theory of 
binary forms will be given later. 



206-207] TERNARY FORMS 255 

The equation of a straight line being 

UlCCi + u^x., + 113X3 = 0, 

Ui, U.2, W3 are as usual called the coordinates of the line. Following 
Clebsch it is preferable to regard the equation just written as 
neither the point equation of the line nor the line equation of the 
point, but as the condition that the line (u^, Uo, u-^ and the point 
(a?!, X2, X3) should bear a certain relation to one another. 

The first fact to notice about (wj, ti^, ih) is that they are 
contragredient to Xi, x.^, X3, for if Ui, u^, u^ become Ui, U.,, U3 and 
x^, x\2, X3 become X^, X^, X3, then 

u■^x■^ + u^x.2-¥ U3X3 must become U^X^-\- U^X^-\- U3X3, 

which is precisely the condition for contragredience (§ 39). 

Further, the us being contragredient to the x's are cogredient 
with the a's, and in fact by the linear transformation of § 201 we 
see at once that Wj becomes u^, it^ becomes u^ and Wg becomes u^. 

It is now evident that by a linear transformation the factor 
(a^u) merely becomes itself multiplied by the determinant of 
transformation, so that every form whose symbolical expression 
consists exclusively of factors of the types 

possesses the property of invariance. 

There are four classes of such invariant functions. 

(i) Containing neither x's nor us. These as already mentioned 
are called invariants. 

(ii) Containing x's but not us. These are covariants. 

(iii) Containing u's but not x's. These are new introductions 
and are called contravariants. 

(iv) Containing both u's and x's. These are called mixed 
concomitants. 

Further there is the identical invariant form u^ which does 
not contain the coeflficients of the original forms at all. 

The degree to which the coefficients occur in a given invariant 
form is called simply its degree, the degree in the x's is called its 
order and the degree in the u's is called its class. 



Chw 


flioi 


U, 


^020 


C^Oll 


M2 


floii 


^^002 


Us 


^<2 


Us 






256 THE ALGEBRA OF INVARIANTS [CH. XII 

Examples of invariants and covariants have already been 
given. 

As an example of a contravariant we may mention 






as a contravariant of the conic. In fact equated to zero it 
represents the line equation, and the reader may verify by direct 
multiplication that it is equivalent to 

(a^uy, 

and therefore actually is an invariant form. 

As an example of calculation similar to that given for the 
Hessian consider the locus of a point whose polar conic with 
respect to the cubic 

touches a given straight line u. 
The polar conic of the point 1/ is 

and the condition that this should touch the given line is 

Thus the equation of the required locus is 

(a/3aycL,^^ = 0. 

This is a mixed concomitant of the cubic. Equated to zero it 
represents the point equation of a curve when the lis are taken as 
constants, viz. the point equation of the locus mentioned. When 
the x's are taken to be constant it represents the line equation of 
a curve, viz. the line equation of the polar conic of the point w. 

Similar remarks apply to mixed concomitants in general. 

208. Principle of Duality. The fact that the condition 
that a straight line whose equation is a^; = should pass through 
a given point is symmetrical in the coordinates of the line and the 
point leads to an important remark. For if the proof of any 
theorem relating to a figure containing m straight lines and n 



207-209] TERNARY FORMS 257 

points is thrown into an analytical form, then by interchanging 
point and line coordinates throughout the investigation we obtain 
a correlative theorem for n straight lines and m points — if in the 
first figure one of the straight lines pass through one of the points, 
then in the second figure the corresponding point lies on the 
corresponding straight line. It must be clearly understood that 
such theorems as are here contemplated depend only on lines 
passing through points and points lying on lines, so that the 
process is exactly analogous to the reciprocation of descriptive 
properties. Naturally we can pass to point loci and line envelopes 
by the introduction of an infinite number of points and lines into 
the figures. 

A fuller explanation of this principle of duality will be found 
in works on geometry, but the above will suffice as an indication 
of some general ideas which will underlie much of our subsequent 
work, especially on the theory of two conies. 

Ex. (i). Shew that in every case the theorem dual to a given one may be 
obtained by reciprocating the given one (supposed descriptive) with respect 
to a conic. 

Ex. (ii). If 2), E, F be three points in the sides BC, CA, AB of a triangle, 
and AD, BE, CF be concurrent, then the lines EF, FD, DE meet BC, CA, 
AB respectively in thi'ee coUinear points. What is the dual theorem ? 

209. Methods for transforming symbolical expressions. 

As in the case of binary forms there are two different methods by 
which a symbolical expression may be transformed : 

(i) the interchange of equivalent symbols, 

(ii) the use of identities among elementary symbolical ex- 
pressions. 

As an illustration of (i) we may notice that for the ternary 
quadratic aj = fi^^ — etc. the mixed concomitant {a0u)ax^x 
vanishes identically. 

In fact, we have 

since a, /8 are equivalent and 

(o^w) ax^x = - i^oLu) ax/3x 
by the properties of determinants, hence (a/3w) ax0x = 0. 

G. & y. 17 



258 



THE ALGEBRA OF INVARIANTS 



[CH. XII 



The fundamental identity for ternary forms is 

(^878) a, - iySa) ^^ + (Ba^) 7, - {ccBy) 8^ = 0,' 

deduced from 

j ai /3i 7i 5i == 0. 

I as ^2 72 S.2 

«3 ^3 73 83 
«« ^X Jx Bx 

On replacing 8 by u 

(^yu) ax - (yua) ^^ + (uafi) yx = {aM "« 

or {^yu)ax + (yau)^x + {ai3u)yx = Wy)Ux, 

an easily remembered result. 

Again, if in the first identity on replacing x by (e^), i.e. 
x^, X2, X3 by €2^3-63^2, ^3^1-61^3, 61^2 -eofi 
respectively, we deduce 

(«^7) (H) - (M) (yeO + (7S«) (0eO - (0y8) (a.n = 0. 
Another important identity is 

where Wj, u^, u^ are the coordinates of the line joining the points 
a; and y so that 

Ui = a^a^s - ^^s^i, W2 = a;3i/i - x^ y^, u^ = x^y.- a?22/i . 

In exactly the same way we have 

where x is the point of intersection of the lines v and w. 
Ex. For the ternary quadratic a^=^^=Qtc. shew that 

(a^u) (ayu) ^^y,=i (a/3«)2 . y,2 _ j („^y)2 . «^2. 

210. We now come to the fundamental theorem in the 
present calculus, namely, that every invariant form can be re- 
presented symbolically by three types of factors, viz. 

(a/37), («/S'0 ^-J^d Oy. 
together with the identical invariant form iix. * 



209-211] 



TERNARY FORMS 



259 



Some preliminary observations will enable us to simplify the 
proof. 

In fact an invariant form containing the u'» may be regarded 
as a pure covariant of the original form or forms together with the 
linear form 

and hence if the theorem be proved for any number of ground 
forms for concomitants in which the us are absent the general 
case follows at once by adjoining a linear form to the original 
system. 

Let us then proceed to prove the theorem for invariants and 
covariants. 



In the proof we 
denoted by 


need 


two lemmas regarding the operator O 




d 

ail 


d 


d 






a 


d 


a 






dvi 


dv. 


dVs 






d 

ari 


d 


d 

ara 





which is the natural extension to three variables of the operator 
so important in the binary theory. 

211. I. // D denote the determinant 

li I. la 
Vi V-2 Vs 

?I ^. ^3 

then H*'!)"' is a numerical multiple of D"-*", where n, r are 
positive integers and r ^ n. 

This result can be proved for r = 1 by straightforward 
differentiation and then the general result follows. We may 
shorten the proof by using properties of the minors of the 
determinant. 

For a moment denote the minor of |i by Xj, that of 7}^ by Fj, 
-of ^1 by Z^, etc. 

17—2 



260 THE ALGEBRA OF INVARIANTS [CH. XII 

Then 
and p^ = n (n - 1) D'-^ F3Z, - nD"-' f ^ . 

0973d^2 



Hence 



B'^^iiin-l) D"^- ( Y.Z, - Y,Z,) + 2nZ)"-i |, 



and Y,Z,-Y,Z,= ^,D 

by the properties of minors, hence 

/ d^ d^ \ _ 

Consequently 
4 (^^ - ^) ^" = ^ (»i + 1) (^ - 1) I)"-^^i^a + n{n + 1) i)"-S 

and adding to this two like terms we have 
nD'' = n(n + l){n-l) D"-^ {^.X, + ^^X, + ^,X,) + 3w (n+ 1) i)"-i 
= n{n+l){n + 2)D''-\ 
since 

Operating with fl again we have 

n^D^ = (n - 1) n^ (w + 1)2 (n + 2) D^-^, 

and proceeding in this way we see that D>^B^ is a numerical 
multiple of D^~^ and, in particular, n**/)" is a numerical constant. 

The reader will find it instructive to extend this property of 
the operator H to four or more variables. 

212. 11. If S be a product of m factors of the type a^, 
n factors of the type ^,, and p factors of the type 7^, then 
iVS is the sum of a number of terms each containing r factors 
of the type {oL^y), m — r factors of the type aj , n — r of the type yS, 
and p — r of the type 7^ . 



211-213] TERNARY* FORMS 261 

The proof of this result is exactly similar to that of the 
corresponding theorem in binary forms. In fact let S = PQR, 
where 

then ^'^ - = y a, w P (") yj^) — _^ -^ 

the sum being taken for 

/Li = l, 2, ... m; 

V = 1, 2, ... n ; 

TV =1,2, ...p. 

Writing down all six such terms and adding we have 

P 

which proves the theorem for r = 1, since — -- contains m — 1 factors, 

and so on. 

Now £1 has no effect on a factor (a^y), hence applying the 
same result to each term in H . >S we obtain the result for r = 2 
and so on for any value of r. In particular if r = m = n=p the 
result is expressed exclusively in terms of factors of the 
type (a^7). 

213. Our proof that any invariant or covariant can be repre- 
sented by factors of the two types 

((X0y), a^ 

follows exactly the lines of the second proof of the corresponding 
theorem for binary forms. As the proof is exactly the same for 
any number of forms, we give it explicitly for one only. 

First suppose that / (a) is an invariant of the form 

nl 
-^^ p ! g- ! r ! 

then if ^Ay^p. — ^ — j — , X-^X.^X^ be the transformed expression, we 

have 

/(^) = /(a).(|i70"' identically. 



262 THE ALGEBRA' OF INVARIANTS [CH. XII 

Now express the left-hand side of this equation in symbols. 
We have 

cipqr = cL^oi2^a/, and hence ^^^ = ixi^a^^a^ *■. 

Thus on the left we have the sum of a number of terms. Each 
contains w factors of the type a^, w of the type a,, and w of the 
type a^. 

Hence operating on both sides of the equation with H^, the left- 
hand side becomes an aggregate of terms each made up entirely 
of factors of the type (a^y) and the right-hand side is a numerical 
multiple of 1(a), so that we have expressed I (a) in the required 
manner. 

The proof for a covariant is similar in form to the above, 
but as in the case of binary forms, a little care is necessary. 
Of course, here as always, we confine ourselves to co variants that 
are homogeneous in the variables x^, x^, x^. 

Suppose that G {a, x) is a covariant of order m, then by 
definition 

C{A,X) = {^ri^YxG{a,x). 

Now on solving the equations of transformation 

a?i = ^1^1 + Vi^^ + ^i-X'3, etc. 
we have 

^' (M) ' 

On replacing Xy, by (^2*^3 - '^3^2), ^^2 by {v^Wi-v^w^) and x^ by 
iy^Wz — v^Wi) — a set of equations which may be written x = {vw) — 
we have 



X,= 



^ ^V^Wf-V^W^ 



W( — V( 



Again, Apqr = a-^°^^o.i, and hence on substituting for the 
A'b and ar's their values as given above in G {A, X), and 
multiplying across by (^0"*> ^^e left-hand side becomes an 
aggregate of products of factors having the suffixes f , ?;, ^, and the 
right-hand side becomes (^t;^)'*^'" x G{a, x). Moreover each term 



213] TERNARY FORMS 263 

on the left-hand side must contain (w + m) factors with each 
siiflSx. 

Now if we operate on the left-hand side once with CI, we 
obtain an aggregate of terms each containing one determinantal 
factor and (tu + ni — l) factors with each suffix. 

The determinantal factors are of three types : 

(i) («yS7), 

(ii) (avw) = a^, 

(iii) (o^/3v), 

where the first two are of the form we require in the result, 
but the third is not. 

Now a term involving a factor (a^v) must have arisen from 
operating with 12 on a term containing a factor such as ol^^^v^ 
and this term must therefore contain a further factor w^ (or w^). 
Accordingly let the term be in the original expression 

then from the mode in which v and w occur in the expression on 
the left-hand side there must also be present a term 

and on operating with H this gives 

whereas the former term is 

But by the fundamental identity 

(a^v) Wr, + i^wv) a, + (wuv) 13^ = (wol/3) Vr, , 
or (a^v) Wr, - (a^w) v, = (fivw) a, - (avw) /3^ = /3x^r, - c^^r, , 
and the sum of the two terms mentioned is 

G{^xOL,-Oix^,). 

Thus although factors like {cl^v) do appear explicitly after 
operating with H, the terms containing them may be paired in 
such a way that the aggregate may be expressed entirely in terms 
of factors of the types (a/Sy) and cix together with factors having the 
suffixes I, 7], or ^. 



264 THE ALGEBRA OF INVARIANTS [CH. XII 

Hence if we operate with H, lu + m times on the left and 
transform at each step as indicated above, we finally have an aggre- 
gate of terms each containing {w + m) factors of the types (a/Sy) 
and Ox only; also after performing the same operations on the 
right-hand side, we are left with a numerical multiple of C (a, x). 
We have therefore expressed the covariant as an aggregate of 
terms each of which is the product of a number of factors of the 
types (a/S7) and a-g, and since the order must be m, it follows 
that in each term there are lu factors of the type (a/87) and m of 
the type a-B. 

214. Leading Coefficients of Concomitants. In § 33 it was 

shewn that a covariant of a binary form could be deduced from its leading 
coefficient; we shall in what follows consider the extension of this idea to 
ternary forms, but as the results are easily obtained and not necessary for 
present purposes our remarks will be somewhat brief ; the reader who 
desires further information is referred to a memoir by Forsyth, Am. Journal 
of Mathematics, 1889. 

If a mixed concomitant be of class m' and order n', the coefficient of 
x-^'u{^' is called the leading coefficient. 

Given the concomitant, the leading coefficient is unique of course, but the 
reverse is not true, because if we multiply a concomitant by any power of Ux 
we obtain another concomitant with the same leading coefficient. 

However, save as to a power oi Ux, a concomitant can be found when its 
leading coefficient is given, as we proceed to shew. 

The leading coefficient is an aggregate of products of factors of the types 

and on replacing a, by a^, 

oj by aj,, 

«3 hy oj, 
and so on, the above factors become 

{a^y)Ux, (a^u), Ox 
respectively, where u={yz). Cf. § 209. 

The symbolical substitution is equivalent to replacing the coefficient 
C'pqr by 



fi(^ 






so that if in a leading coefficient we replace Op,,. by the value just given we 
obtain the corresponding concomitant multiplied by a power of Ux — the ex- 
ponent of Ux is in fact equal to the number of factors of the type {afiy) that 
occur in the symbolical expression of the concomitant. 



213-215] TERNARY FORMS 265 

Hence, as in binary forms, it follows that " If there be an identical relation 
between a number of leading coefficients, the same relation exists among the 
corresponding concomitants." 

We leave the reader to establish this on the lines of § 33 ; the con- 
comitants corresponding to the various leading coefficients must be chosen 
so as to make the whole expression of uniform order and class. 

On multiplication by a sufficiently high power of Ux a concomitant can be 
completely expressed by means of factors of the types (a^«) and a*. 

The leading coefficient can therefore be expressed in terms of factors of 
the types 

("2^3-03^2) and oi. 

It follows immediately that the leading coefficient is an invariant of the 
binary forms 

(02.^2 + 03^3)", (02^72 + 03.^3)" -ifli, {a,^X^_^->r a^X^ a^-\ 

which are n in number. 

These binary forms are the coefficients of the various powers of x^ in the 
original ternary form, hence they can be obtained at once from /. 

This is the fundamental result of Forsyth on algebraically complete sets 
of ternariants and from it readily flow all the results in the memoir cited. 
Similar methods may be applied to obtain the results of another paper of 
Forsyth's*. 

215. We shall now explain an important principle due to 
Clebsch which establishes a connection between the invariants of 
binary forms and the contravariants of ternary forms. To facilitate 
the discussion we shall commence with a special case. 

Suppose we require to find the condition that the line Ur^ — 
should touch the conic a^- = h^- = 0. Let the points on the line be 
yxViVz ^■iid z-^z-^z^, so that 

Wi : u^ : M3 = ijoZ^ - y^Zo : y^z^ - y^z^ : y^z. - y^z^. 

The coordinates x^, x^, x^ of any point on this line are of the 
form 

Il2/l + l2^1> |l2/2+fe, I12/3 + 12-^3, 

and fi, ^2 may clearly be regarded as the coordinates of a variable 
point on the line referred to the points y and z as base points. 

We have 

* See London Math. Soc. Proc, 1898. 



266 THE ALGEBRA OF INVARIANTS [CH. XII 

and the points in which the line meets the conic are given by 
(aj/fi + a,^,y = {hy^, + hU = . . . = 0. 

As the line touches the conic this expression regarded as a 
quadratic a^^ = ^^^= ... in ^ must have its invariant (a/3)- zero. 

Hence {aybz — azbyf = 0, 

and by an identity already given 

{Oyhz — agby) = (abu) ; 

hence we have {abuf = 

as the required condition. 

Ex. If tike line ttg = cuts the conies ax^ = 0, h:^=0 in harmonic points, 
prove that {ahuf=0. 

Thus when the points on the line satisfy the invariant relation 
{cL^y = 0, the line satisfies the relation {ahuf = 0. 

This principle is true in general, that is to say, in order to 
pass from the invariant relation satisfied by the points in which 
the line meets any number of curves to the condition satisfied by 
the coordinates of the line, we have merely to change every factor 
of the type (a/3), in the expression of the binary invariant, into a 
factor {ahu). 

The proof is very simple — suppose the equations of the curves 

are 

a^'« = 0, K'' = 0, etc., 

and that y, z are two points on the line, then the points in which 
the line meets the curves are given by the binary forms 

(ayfi + a^^,)^ = 0, {hy^, + h^,Y = 0, etc. 

But an invariant of these forms is expressible in terms of 

factors of the type 

{ayh^ - azby) 
entirely, and inasmuch as 

a-ybz - ttzby = (ahu) 

the result follows at once. 

Thus, knowing the discriminant of a binary form of order n, we 
can find the tangential equation of a curve of the nth degree, for 
the discriminant being equated to zero gives the condition that 



215-217] TERNARY FORMS 267 

two points of ntersection should coincide, and in this case the 
line touches the curve ; e.g. the discriminant of a binary cubic 
being 

{ahf (ac) (bd) (cdf, 

the tangential equation of the curve 

dx^ = hj = Ca;^ = dx^={) 

is {ahuf (acu) (bdu) {cduf = 0. 

216. The principle can be extended to covariants and mixed 
concomitants, as we shall explain by an example. 

A line Ma; = cuts two conies a^^ = 0, bx^ = in pairs of points 
PQ, RS: to find the pair of points harmonic to both P, Q and 
R, S. 

We know that the pair of points harmonic to those given by 
af2 = 0, 0f = 0, 
are given by equating the Jacobian (a/S) a^ yS^ to zero. 

Now if 2i be the join of the points y and z, then fi, ^2 are given by 
(a,^i + ch^.Y = 0, {by^, + bSy = 0, 
hence the common harmonic pair are given by 

(ayb^ - Uzby) (ay^i + a^^a) (by^i + h^^) = 0, 
or they satisfy (abu) Uxbx = 0, 

and are therefore the points in which this conic meets the given 
line. 

The extension to covariants in general will now be sufficiently 
obvious. 

217. There are also dual methods which enable us to deter- 
mine, for example, the locus of a point such that the pencils of 
tangents drawn from it to a number of fixed curves possess a 
given projective property. 

After the study of a simple example it will be easy for the 
reader to enunciate and prove the necessary theorems. 

Let us find the locus of a point such that the pairs of tangents 
drawn from it to two conies 

Ua' = 0, u^^ = 0, 
are harmonically conjugate. 



268 THE ALGEBRA OF INVARIANTS [CH. XII 

If any two lines through the point are v^ =0, Wx = 0, then the 
coordinates of a variable line through it are 

where |i, ^2 may be regarded as the coordinates of a variable line 
in the pencil. 

The tangents to the two conies are given by the binary 
quadratics 

(vaai+u^a-j + Usa^y, or (Wa|i + Wa|2)' = 0, 

and (vp^i + 'i^p^2y = 0. 

And since these are harmonic 

{VaWp — V^Way=0, 
bu t VaWfi — V^ Wa = {OL^X), 

where x is the point of intersection of the lines v and w. 
Hence the required locus is 

{a^xf = 0. . 

If a and y8 are equivalent symbols, this gives the point equation 
of the conic whose tangential equation is 

W„2 = w^2 = 0. 

Ex. (i). Prove that if three binary quadratics a^^, h^, c^ are in involution, 
then {ab) (be) (ca) = ; deduce the locus of a point such that the tangents from 
it to three conies form a pencil in involution, and state the correlative 
theorem. 

Ex. (ii). Interpret the envelope loci {abu)* = 0, {abuf'{hcuf{caiCf = in 
connection with the quartic curve ax* = bx* = Cx* = 0; shew that the inflexional 
tangents touch each of the above curves, and are therefore in general com- 
pletely determined as the common tangents. 

Ex. (iii). A straight line p meets a cubic curve in P, Q, R, shew that 

(i) there are two points IT, W on the line such that their polar conies 
touch the line ; 

(ii) the polar conic of H touches the line in H' and that of E' touches 
the line in H ; 

(iii) H and H' are the Hessian points of P, Q, R; 

(iv) if p pass through a fixed point y the locus of H and H' is the 
quartic obtained by putting u={xy) in {abuy axhx'^0 ; 

(v) the equation of the quartic is fP-C'^ = 0, where /=0 is the 
cubic, C the first and P the second polar of y. 



217-218] TERNARY FORMS 269 

218. Quaternary Forms. The general quaternary form of 
order n is 

plqlrl si 
where , p + q + r + s = n. 

In accordance with the methods used in this work, we write 
this in the umbral form 

and this in turn we denote by ax\ 

To represent expressions of degree higher than unity in the 
coefficients, we have to introduce equivalent symbols yS, y, etc., 
so that 

/=a/^ = /3^" = 7^'' = etc. 

Denoting the determinant 



«! 


a., 


O-i 


0-i 


^1 


/3. 


/33 


/34 


7i 


72 


73 


74 


B^ 


8, 


K 


^4 



by (a/SyS) and extending the definitions of invariants and co- 
variants, we can shew that every expression completely represented 
by factors of the type (a^jB) is an invariant, and every expression 
represented by factors of the types (a^yS) and ax is a covariant ; 
of course in a symbolical expression each symbol must occur to 
degree n. 

Then using plane coordinates u, such that 
UiXi + U2X2 + U3X3 + iiiX^ = 

is the equation of the plane ii, we have mixed concomitants 
containing the three types of factors 

ax, (a^78), (a^yu). 

Finally, introducing a second plane v, we have a more compre- 
hensive type of mixed concomitant expressed by factors of the 
four types 

ax, (a^7v), (a^yu), (a^uv). 

There is no need to introduce a third set of plane coordinates w, 

because 

{auvw) 

/ 



270 THE ALGEBRA OF IXVARIANTS [CH. XII 

is really of the form ax, where x is the point of intersection of the 
planes u, v, lu. 

The reader will not have more than purely algebraical diffi- 
culties in extending the methods of §§ 211, 212 to four variables. 

219. In illustration of the foregoing we may mention that if 

/ = a^' = ^x^ = li = ^x 
be a quaternary quadratic, then 

(i) {oL^^hy^ is its discriminant ; * 

(ii) {a^>yuf = is the condition that the plane u should 
touch the quadric /= ; 

(iii) {oL^uvf = is the condition that the line of intersection 
of the planes u and v should touch the quadric ; 

(iv) (auvwY = is the condition that the point of intersection 
of the three planes u, v, and w should be on the quadric. 

220. It is clear that when any invariant form expresses a 
relation of a projective nature between a straight line in space 
and a surface, factors of the type 

(a^uv) 

must occur in its symbolical form, because the straight line is 
given in the first place as the intersection of two planes, which 
are u and v in this case. 

It is easy to see that 

(a^UV) = S (ttiySa - ftzySi) (1*3^4 - W4V3) ', 

the six quantities 

MjVg — W3V2, UsVi — UjV-i, KtV-^ — U^Vi, UiVi-tUVi, U^Vi — UtVi, UsVi-UiVs 

are called the six coordinates of the line of intersection of the 
planes u and v. It can be verified that they are altered in the 
same ratio, when instead of the planes u, v we take any other two 
planes through the line. 

Further, if cc, y be any two points on the line, the quantities • 

a;i3/4-^42/i. Xc^A-Xiy^, x^Vi-oc^z, ic^yz-x^y^, ^syi-^iya, ^12/2-^22/1 



«1 


Xn 


a?3 


«4 


2/1 


2/2 


2/3 


2/4 


Wl 


W2 


M3 


%4 


Vi 


^2 


V3 


^4 



218-221] TERNARY FORMS 271 

are altered all in the same ratio, when instead of x, y we use two 
other points on the same line. 

Finally, from the equations 

UX = 0, Uy = 0, Vx= 0, Vy = 0, 

it follows that 

^l2/4 -X^^ = p (U.V3 - U3V2) =pu, 
^22/4 - a?4y2 = P (W3V1 - W1W3) =^24, 
.-^^32/4 - «42/3 = P ("1V2 - M2V1) = i>34, 
^22/3 - ^32/2 = P (^^^4 - W4V1) = P23 , 
^3^1 - ^l2/3 = /3 (^<2'y4 - ^4^2) =P31, 
^l2/2 - ^22/1 = P (^^4 - W4V3) = ^12 , 

in other words, that the six coordinates of a line are either the six 
determinants of the array 



or those of the array 



The expression (a^iw) is written (a^p) for convenience, but 
it must not be confused with the similar expression for a three- 
rowed determinant. 

The discussion of concomitants involving line coordinates is 
complicated by the existence of the relation 

PtsPu + P31P24. + PiiPsi = 
between the six coordinates of a line. 

221. As a simple illustration of the use of the above methods, 
we shall take the extension of the principle of Clebsch explained 
in § 215. 

To explain this extension we take a particular problem, viz. 
to find the condition that the plane Wa, = should touch the 
quadric surface represented by 

(^x- = yS^' = jx- = 0. 
Let X, y, z be three points in the plane, then any other point 
(Xi, X2, X3, X4) in the plane is given by 

Xr = f 1.^. + t^Jr + Ur, ^ = 1 , 2, 3, 4, 



272 THE ALGEBRA OF INVARIANTS [CH. XII 

and ^1, ^2, ^3 may be regarded as the coordinates of (X^, X^yXs, X^) 
in the plane referred to x, y, z as fundamental triangle. 

4 
Now 'la.rXr = aa;f 1 + O^^s + fl^^s , 

1 

hence the equation of the conic in which the plane meets the 
quadric may be written 

If the plane touch the quadric, this conic must be a pair of 
straight lines ; the condition for this is 



O-x ««/ ftz 
^x ^y ^z 
1% ly Iz 



= 0, 



and inasmuch as the determinant here written is equal to 

the condition required is 

(a/37M)2 = 0. 

The reader will have no difficulty in applying the same 
method to find the envelope of a plane cutting a surface in a 
curve which has a definite projective property, of which the 
invariant equivalent is given. 

222. We can in like manner solve problems leading to line 
coordinates, e.g. to find the condition that a line should cut the 
quadrics 

a^' = 0. ^9x^ = 0, 

in pairs of points harmonically conjugate. 

Let X, y be two points on the line, then any other point X 
is given by 

Xr = ^lOCr + ^23/r, ^ = 1, 2, 3, 4, 

and ax = cl^^i + Oy^i, 

so that the quadratics giving the ratio in which the line is divided 
by the quadrics are 



221-222] TERNARY FORMS 273 

The condition that the pairs should be harmonically conjugate is 

{^x^y - tty^xf = 0, 

or (a/3py = 0, 

where p is a typical coordinate of the line. 

The construction of other examples and the discovery of the 
dual principles will be easy for the reader who has grasped the 
corresponding results relating to ternary forms. 



G. & Y. 18 



L 



CHAPTER XIII. 

TERNARY FORMS (continued). 

223. In this chapter we shall give some further theorems 
relating to ternary forms; mainly such as arise in obtaining the 
irreducible systems in the simpler cases. In connection with 
Hilbert's proof of Gordan's theorem it has already been remarked 
that this proof can be extended without much difficulty to the 
case of ternary forms; we are therefore certain that for any number 
of such forms the irreducible system is finite, i.e. there exists a 
finite number of invariant forms in terms of which all others can 
be expressed as rational integral algebraic functions. But although 
the existence of the finite system is thus established, no clue 
is. given as to the method of discovering such systems and, as a 
matter of fact, very little is known in this branch of the subject. 
The more important complete systems known up to the present 
are those for one, two* and three f ternary quadratics, that for a 
single ternary cubicj form, and quite recently Gordan has given the 
complete system for two quaternary quadratics§. 

224. In his paper on the ternary cubic Gordan gave a syste- 
matic method of searching for all irreducible forms by proceeding 
from those of degree r — 1 to those of degree r in the manner of 
Chapter v. ; in any but the simplest cases the application of this 
method requires uncommon skill and patience. The reader will 
find an introductory sketch of the method at the end of this 
chapter ; we shall content ourselves with obtaining the complete 
systems for one and two quadratics by an elegant and ingenious 

* Clebsch, Geometric, p. 174. 
t Ciamberlini, Battaglini, Vol. xxiv. 
X Gordan, Math. Ann., Band i. 
§ Math. Ann., Band lvi. 



i 



223-225] TERNARY FORMS 275 

process, also due to Gordan, which has been applied with success 
to three, and would probably be equally successful in dealing with 
any number of, quadratics. The complete systems are in all cases, 
except for linear forms or one quadratic form, so large that any 
method for their discovery will involve a great deal of labour. 

Ex. (i). For a single linear form a^ the complete system is «„. 
Ex. (ii). For two linear forms a^ and b^ the complete system is 

a^, b^, {abu). 

Ex. (iii). For any number of linear forms a^, b^, c^, ... the complete 
system consists of 

(a) Forms of the type %, 

(/3) Contravariants of the type (abu), 

(y) Invariants of the type (aba). 

These results all follow immediately from the symbolical expressions for 
invariant forms. Cf. § 84. 

225. One Quadratic Form. Let the form be 
f^ai = h^ = Co? = etc., 
then the t3rpical invariant factors are 

(abc), (abu), a^, 

and there are three invariant forms 

(abcf, {abuf, a^ ; 

we proceed to shew that these constitute the complete irreducible 
system. 

I. Any invariant form P containing the factor (abc) can be 
transformed so as to contain the factor (abcy. 

First suppose that no two of the letters a, b, c occur again in 
the same factor, then 

P = (abc) apbqCr M, 

where M does not contain a, b or c, and 

p = (de), (dib) or x etc. 

Hence by interchanging a, b, c in all possible ways 

6P = (abc) M(apbqCr — apbrCq — aqbpCr + ttqbrCp — arbqCp + arbpCq) 

— (abc) M(abc) (pqr), 
= (abcf M (pqr). 

18—2 



276 THE ALGEBRA OF INVARIANTS [CH. XIII 

Thus P involves the factor {abc)- which is an invariant, hence 
the other factor must be an invariant form* and P is therefore 
reducible. 

Secondly, suppose that 

P = (abc) (abp) CqN, 

where p = d or u, 

q = (de), (du) or x. 

Then 3P = (abc) iV {(abp) Cq + (bcp) a^ + {cap) bq] 

= {abc) N {abc) pq 

= {abcyNpq, 

and as before it follows that P is reducible. 

Thus any form containing the factor {abc) is reducible, and we 
may neglect such forms in future. 

. II. A form Q containing the factor (abu) can be expressed in 
terms of forms containing the factor {abuf and reducible forms. 

Rejecting forms containing factors of the type {cde), we have 
the three following possibilities : 

(i) Q = {abu) {acu) {bdu) M, 

(ii) Q = {abu) {acu) b^ M, 

(iii) Q = {abu)aJ)xM. 

As regards (i), on interchanging a and b, 

2Q = {abu) M [{acu) (bdu) + {cbu) {adu)] 

= {abu) M j — {bau) {cdu)] 

= {abuy {cdu) M, 

and Q is reducible. 

(ii) Here 

2Q = {abu) M [{am) b^ - {bcu) a«} 

= {abu) M [{ahu) Cx — {abc) Ux] 

= {abuf Mcx — {abc) u^ {abu) M, 

and the latter part is reducible because it contains the factor {abc). 
Thus again Q is reducible. 

* We shall return to this point later, see § 226. 



225-226] TERNARY FORMS 277 

(iii) In this case Q vanishes, as we see by interchanging 
a and b. 

Now every symbolical expression representing an invariant 
form contains a factor of one of the types (abc) or (abu) unless 
it be ajT] it follows that all such are reducible except 

(abcy, (abuY, a^-, 

and these form the complete irreducible system. 

Of course, to be strictly accurate we should add to this and 
every complete system the identical concomitant u^. 

Ex. (i). Express (abu) (acu) b^c^ in terms of the irreducible system. 
Ex. (ii). Prove the symbolical identity 

{ab (pq)} = apb, - aj)p . 

226. The foregoing discussion leads to some general remarks 
that will be useful in the sequel. 

The artifice of replacing {abc) by cip, or, what is the same thing, 
writing 

{bc)=p, 
and more explicitly 

(62C3 - 63C2) =Pi ; (&3C1 - 61C3) =^2 ; {biCi - b^Ci) = ps, 

will be frequently used. (It is clear that if b and c be two straight 
lines then p is their point of intersection.) 

In particular the contravariant {abuf will be written u„^, thus 

a = {ab\ 

and the invariant of the quadratic is Ca. 

The a's may be regarded as expressible in terms of the original 
symbols, but we shall more often regard them as independent 
symbols ; the contravariant above will thus be 



2_ „,2_^ o, 2^ 



the original form is 

J ^ tt/g" ^ Of' ^ Cat" = . . . , 

and the invariant is 

«a" = iifi^ = b^- = etc. 

Of course in replacing an a by symbols belonging to /, we 
must use no symbol already occurring in the expression because 



278 THE ALGEBRA OF INVARIANTS [CH. XIII 

then that symbol would occur more than twice in the transformed 
expression. 

The a's being formed from the a's and &'s, as line coordinates 
are formed from point coordinates, it follows that the a's are 
contragredient to a and u but cogi'edient with x. 

We shall now return to a point that arose in the discussion of 
a single quadratic, where we deduced from first principles that if 
Op, ttq, ar be all invariant factors then {^qf) is an invariant factor 
— it will be shewn here how to express it in terms of factors of the 
types {ahc), (abu) and a^. 

The possible forms for p, q, r are (de), (du) and cc ; if each 
of them were cc, (pqr) would vanish, so we may suppose that p for 
example is not x. 

Let p = (dv), where v is either e or u, then 

{pqr)={{dv)qr}=dqVr-drVq, (§ 209) 

but since bq is an invariant factor, dq and Uq are both invariant 
factors ; for if in such a factor we replace a letter of the type a 
by another of the same type the factor is still invariant, while if 
we replace it by it the resulting factor either vanishes or is an 
invariant factor. 

Hence (pqr) can be. expressed entirely in terms of factors of 
the types (abc), (abu) and a^ ; it is therefore an invariant factor. 

227. To make the matter clearer we shall illustrate it by 
some examples. 

Suppose a quadratic is 

U/X — ^X — ^X — • • • > 

and its contravariant is 

then tta, ap,ba, bp etc. are invariant factors. 

Hence (a fix) must be an invariant factor and (a/Bxy an in- 
variant form of the quadratic. We proceed to express it in terms 
of the members of the irreducible system as follows : 

(afixy = (ab fixf = (apb^-ax bpf 

= a^^ . b^ + a^ . b^ - la^b^a^b^, 
and we have now to deal with the term Uxbxapbp. 



226-227] TERNARY FORMS 279 

As a guide to the further reduction we remark that since a^ is 
a factor the form must involve the factor a^^ after suitable trans- 
formation. 

Thus 

ajy^a^h^ = a^bx (acd) (bed) = (bed) ax (acd) bx, 

and Saxbxa^bfi = (bed) ax {(acd) bx — (abd) Cx — (acb) dx}, 

or since (acd) bx — (abd) Cx — (acb) dx = «« (bed), 

Sttx bx ap bp = (bcdy a^. 
Consequently 

(a^xf = 1a^ . bx' -^aj. bx' = ^a,' . bx', 

and except for a numerical factor the invariant form is equal to 
the product of the original form and its invariant. 

The following example will shew the advantage of introducing 
symbols like a, /8, 7 above in geometrical investigations. 

Let ax =0,bx = 0,Cx = be the equations of three straight lines, 
then if a = (be), /3 = (ca), 7 = (ab), 

(^y) = (abc) a. 

This is not surprising, because a, ^, 7 are the points of inter- 
section of b, c; c, a; a, b respectively and (^y) must therefore 
give the line joining ^ and 7, that is a*. 

The equation of the straight line joining the point y to the 
point of intersection of ax = and 6^ = is 

(ab, xi/) = or axby — aybx = 0, 
which is the well-known form. 

The line joining the point (a, b) to the point (c, d) is 
(ab, cd, x) = 0, or (acd) bx — (bed) ax = 0. 

This may also be written (abc) dx — (abd) Cx = 0, and the two 
forms are equivalent in virtue of the fundamental identity from 
which both can, in fact, be at once inferred. 

Now let a', b', d be the sides of a second triangle and a', ^ , y' 
its angular points. 

* The similarity of these ideas to the methods of Grassmana's^iisdfift/mjjgfs/e/ire 
will not escape the notice of the reader who is acquainted with the Calculus. 



280 THE ALGEBRA OF INVARIANTS [CH. XIII 

The lines joining corresponding vertices of the two triangles 
are (aa), (yS/3'), (77'). and these are concurrent if 

i.e. if (a^/3') (a'77') - (a'/3/3') (077') = 0. 

But since (ayS) = (abc) c etc. this condition is 

(abc) {a'h'c') [c^'h'y-c'^hy,] = 0, 

or {abc) {a'b'c') [{a'c'c) {abb') - {ace') {a'b'b)} = 0, 

or {abc) {a'b'c') [{abb') {a'cc') - {ace') {a'bb')] = 0, 

or {abc) {a'b'c') (a^', W, ~^) = 0. 

Now {aa', bb', cc') = is the condition that the points of inter- 
section of corresponding sides should be collinear. 

Hence when the joins of corresponding vertices ate concurrent 
the intersections of corresponding sides are collinear — a well-known 
theorem. A direct analytical proof by the ordinary methods 
without using a particular triangle of reference is by no means 
easy. 

Ex, (i). Prove that the cross ratio of the range in which Ux=0 is met by 
ax=0, bx = 0, Cx=0, dx=0 is — • \\ ^ i i hence the general equation of a 

conic touching the four hues is 

(abu) (cdu) + X {bcu) {adu)=0. 
If the six hnes a, b, c, d, e, f touch a conic, then 

(abe) (ede) (bcf) (adf) = {bee) (ode) (abf) (cdf). 
Ex. (ii). State and prove the results dual to those of (i). 
Ex. (iii). If the points a, /3, y, 8, e, f be on a conic and 

{a^)=p, m = q, (yS) = r, (8e)=y, «) = ?', {Co) = r\ 
then {pp', qq', rr')=0. 

Deduce Pascal's theorem and prove Brianchon's theorem in the same way. 

228. Two Quadratics. Suppose the quadratics are 

y = aJ' = bx^ = Cx^ = etc., 
/' = aV = 6V = cV = etc. 
and that their contravariants are 

(f) = uj = n^ = u.^ = etc. 

^ = uj' = Ufi'^ = Wy2 = etc. 
respectively, then their invariants are aa^ and a ^^ respectively. 



227-228] TERNARY FORMS 281 

The types of invariant factors are 

(abc) = tta, (aba) = a„, (ab'c) = «„', 

(a'b'c) = a'af, (abu) = »„, (a'b'u) = Ua', 

(aa'u), Ux, a'x, 

and in the course of the work we shall use the additional types 

{a^x), (a'/3'a)), (aa'x) 

which, as has been pointed out, certainly are invariant factors. 

There are four types of symbols occurring in any expression, 
viz. 

a, a', a, a', 

and the leading idea of the investigation is to reduce the 
number of symbols in the expression for any invariant form to a 
minimum. 

Thus {abuy involves two symbols a, b, but it can be written 
Ua^ and then only involves one. 

Again 2a„,'b„.'axbx 

is equivalent to {aa'bx — cixb„,)^ 

except for reducible terms ; then since 

aa'bx— Oxbo: = (oLOL x) 

we see that aa.'ba,'axbx 

can be expressed in terms of reducible forms and forms containing 
a smaller number of symbols. 

This example will make clear what we mean in the sequel by 
reducing the number of symbols. 

I. // a symbolical product contain a factor of the type a^ or 
a' a' it is reducible. 

For let P = (abc) ttphqCrM, 

then, as in § 225, 

eP = (abcy.(pqr)M, 

and ( pqr) is an invariant factor ; hence P is reducible. 

This shews incidentally how factors of the type (a^x) arise 
naturally in the course of the work, for we might have 
p = a, q = ^, r = x. 



282 THE ALGEBRA OF INVARIANTS [CH. XIII 

The case in which a and h occur again in the same factor, e.g. 
(abu), is treated as in § 225. 

II. If a symbolical product contain a factor of the type {ahv), 
where v is of the type a' or is u, then it can be so transformed as to 
contain an additional factor (abw). 

Suppose in fact that 

F = {abv)apbqM, 

then 2P = (abv) (apbq — aqbp) M 

= (abv) {ah, pq) M 

= Va. (otpq) M, 

and (apq) can be expressed in terms of a and the symbols occurring 
in p and q. 

Hence in the transformed expression there are two factors 
involving the symbol a. 

A like theorem applies to a', and here we see a great advantage 
in introducing these symbols, because just as when one a occurs in 
a product there must be another, so when a occurs the expression 
can be transformed so as to be of degree two in a. 

III. If an expression contain a factor of the type (a^y) or 
{a'^'y) it is reducible. 

For let 

P = (a^y) M, where y must be of the type a, a' or x. 

Then since 

(a^y) = (ab, ^y) = a^by-b^ay, 

P = a^byM-bfiayM 

and by I. the form P contains the invariant a^- as a factor. 

Summing up these results we infer that any concomitant 
other than a^^, aV, ctaS a'a^ must be composed of factors of the 
types 

««. a'sc. c^a', a'o, Ua, Ua', {olol'x), (aa'u), 

and if one a occurs there must be another present after a suitable 
transformation. 



228] TERNARY FORMS 283 

IV. Any expression containing two equivalent symbols can he 
expressed in terms of concomitants that are either reducible or 
contain a smaller number of symbols. 

There are two cases according as the equivalent symbols are of 
the type a or a, so that we have to consider the expressions 

apaqbrbgM and ■sr^paa'pTpM, 

with the exactly similar ones 

a'pa'qb'rb'gM and ■uTa'paO-p'r^'M. 

Since a^ br — ar bp = (apr), 

and ttg bg — agbq = (aqs), 

it follows that expressions derived from 

apaqbrbgM, 

by permuting the symbols p, q, r, s, differ from the original ex- 
pression by forms in which the two symbols a and b are replaced 
by the simple symbol a. 

Hence if any one of the three expressions 

apaqbrbgM, aparbqbgM, apagbqbrM 

can be expressed in terms of reducible forms and forms containing 
fewer symbols the same is true of the other two. 

The same result can be easily established for the three 
expressions 

-^aPaO-^T^M, 'UTa.ar^p^T^M, 't^a.Ta P^CT ^M ] 

in fact the difference between any two is reducible by III. 

Consider now the expression 

apaqbrbgM. 

There are only five possible types for p, q, r, s, viz. a, a, (bu), (a'u) 
and X ; of these a may be neglected because if it occur the form 
is reducible, and (bu) may be neglected because if it occur we can 
replace a and b at once by the single symbol a. 

Hence there are only three remaining possibilities, viz. a', (a'u), 
X, and oi p, q, r, s some two must certainly be of the same type. 

First suppose that two are identical, say p and r, then 
apUrbqbgM^ a.^ . bqbgM ; 



284 THE ALGEBRA OF INVARIANTS [CH. XIII 

thus aparhqbgM is reducible, and hence cipaqbrbsM can be expressed 
in terms of reducible forms and forms containing fewer symbols. 

Similar reasoning applies to the expression 

when any two of the symbols w, p, a, r are identical ; here, as in 
the other case, there are three possibilities, viz. a, a' and u, so some 
two must be of the same type. 

Proceed now to the general case in which no two of the letters 
p, q, r, s are identical. 

In this case some two must be equivalent without being 
identical. 

Suppose the equivalent symbols be p and r, then there are 
three cases to consider. 

(i) Let p = r = x. Here p and r are identical and the result 
has been established already. 

(ii) Let p — OL, T — fi\ then 

ftp ar bq bg M = tta' cip' bq bg M. 
And since the symbols a, ^ each occur once the expression 
can be transformed by II. so that each occurs twice, hence 

aparbqbgM= a^' a^' ja- 8^' ^. 
But aa-ap'ja'B^'N falls under that class of expression 

'STa'Pa.XTp'Tp'M 

in which two of the symbols bt, p, a; t are identical, for -07 = a- = a ; 
hence, as we have seen, it can be expressed in terms of simpler 
forms. 

Thus QpttrbqbgM can be expressed in terms of forms that are 
either reducible or contain fewer symbols, and the same is true of 
apaqbrbgM. 

(iii) Let p = (a'u), r = (b'u), then 

ttpQrbqbgM = (aa'u) (ab'u) bqbgM. 
And the latter expression can be written 

{a'au) (b'au) a't^b'^M', 
or a'p'b'ya'q'b'g'M', 

where p = (au), r' = (au), i.e. p' and r' are identical. 



228] TERNARY FORMS 285 

Thus, as we have ah-eady shewn 

a'p'a'q'h'r'h'^M' 

can be expressed in terms of simpler forms ; hence apUrbqlgM 
can be expressed in terms of forms that are either reducible or 
contain fewer symbols, and the same is true of apaqbrbgM. 

This completely establishes the Theorem IV. when the equi- 
valent symbols are of the type a or a. 

We have still to consider the general expression 

where -ot, p, a, r are each of the types a', (ax) or u. The case in 
which some two are identical has already been discussed and 
reasoning exactly similar to that of (i), (ii), (iii) above enables us 
to prove Theorem IV. for the general expression. 

The theorem we have just proved simplifies vastly the 
evolution of the irreducible system inasmuch as it shews that 
in an irreducible form there cannot be more than one symbol 
of any of the types 

a, a', a, a.'. 

A further limitation is imposed by the following theorem. 

V. A form containing hoth the factors (aa'u) and (aa' cc) may 
be rejected in constructing the irreducible system. 

In fact by direct multiplication 

(aa'u) X {aoLx) = I a„ a'^ Ua 

a^f a a' Ug^ 

I ax a X Ux 

hence if a form involve both these factors it can be transformed so 
as to only contain the simpler factors 

tta, a' a', a' a, ^a', Ua, Ha', ««. d'T ^nd Ux- 

With the aid of the above five theorems the problem of the 
complete system is reduced to a very simple one ; in fact we have 
only to write down such products of factors of the types 

tta, a'o', da', a a, Ua, Ma', ««;, Cl'x, (tta'u), {aCLX), 

as satisfy the conditions implied in the theorems. 



286 THE ALGEBRA OF INVARIANTS [CH. XHI 

Since no two equivalent symbols can occur in the same product 
we need introduce no more symbols ; further a^, a'a- can only occur 
in the invariants aa^ and a'a'^ respectively ; therefore it only remains 
to write down all products of the factors 

da', of a, Ua, Ua', tt^, tt'a;, (tta'u), (au'w), 

in which every letter except u and x which appears at all appears 
twice and no more ; besides the two last factors must not appear 
in the same product. 

Hence we have the following forms : , 

(A) aa\ aV, «»'', a'a^ u^, w.-^, a^, aV, {av^uf, {(idxf, 
obtained by squaring each factor. 

(B) aa'ttajWa', aa'ax{oLOLx)Ua., aa'ax(aax)a'aa'a;, 
ac,'{aa'u)a'x, aa.'{aa'u)a'aUaUa', 

containing the factor a^'. 

(C) a'aa'xUc,, a'aa'x{o.OLx)ua', a'a{aa'u)ax, 
containing a a but not tta-. 

(D) Ua{a.OLX)Ua,>, 

containing Ua but neither a\ nor a^'. 

(E) ax{aa'u)a'x, 
containing a^ but neither a'^ nor a^'. 

There are twenty forms in all, viz.: 
Four Invariants, 

Four Covariants, 

ax^ = Si, a'x^ = 8^, {oLOLxy = F, {ao^x)ao:a\axa'x. 
Four Contravariants, 

tij = 1i, Ua'^ = l2> {aa'uf = ^, (aau)aa'a'aUaUa'. 
Eight Mixed Concomitants, 

cixda'Ua', a'xa\Ua, {aa'u)axa'x, {aa'u) aa-a xU^.; 
{aa'u)aaaxUa, (aa'x)Uaaa'ax, (aa! x)Ua' a' aa'x, 

{aOLX) U^Ua.: 

To these twenty should be added the identical concomitant %. 



228-229] TERNARY FORMS 287 

It should be noticed in conclusion that, although it follows 
from the five theorems of this article that every other invariant 
form is expressible in terms of these as a rational integral algebraic 
function, it has not been shewn that these twenty forms are 
themselves irreducible. Cf note, p. 131. 

Ex. (i). Prove that 
Ex. (ii). Prove that 

and thence that (««', aa')2 = |^Jjjj4222 + -^n2-^i22' 

Ex. (iii). Prove that 

Ex. (iv). Prove that 

a„u.a^(aa'x)=0. 

a a X ^ / 

229. A complete irreducible system of invariant forms may 
be regarded as giving rise to two inquiries. 

(i) What is the geometrical meaning of each member of the 
irreducible system ? 

(ii) What is the expression in terms of the members of that 
system of the invariant forms which arises in the analytical 
treatment of a given problem ? 

To the first of these inquiries an answer can generally be 
given, provided a sufficiently complex geometrical apparatus be 
allowed, but it commonly happens that the significance of some 
members of the system is so remote as to render them of little 
geometrical importance. 

The second inquiry is naturally unanswerable until the pro- 
blem be named, and thus all we can do is to illustrate it by the 
discussion of some simple problems. 

Before going further it may be well to add that the second 
inquiry is the really important one ; in a manner it includes the 
first as a particular case, and in fact there being no direct method 
of proceeding from the invariant to the geometrical meaning, the 
answer of the first inquiry is obtained fortuitously in pursuing 
the second. If it be not obtained we should console ourselves 
with the reflexion that the uninterpreted forms are of little geo- 
metrical interest in the present state of knowledge ; besides, if we 



288 THE ALGEBRA OF INVARIANTS [CH XIII 

regard the algebra as being merely helpful to geometry in the 
analytical formulation of results, it does not follow that everything 
in the algebra need be taken seriously from the geometrical point 
of view*. 

In spite of what we have said, we shall begin the geometrical 
theory of two conies by interpreting the members of the irre- 
ducible system ; it will be seen that they are all of importance in 
elementary geometry. 

230. Geometrical Theory of tw^o Quadratics. The 
irreducible system. The forms themselves when equated to 
zero represent two conies, viz. 

a«=' = 0, aV = 0, 

which we write Si = 0, ^2 = 0, and call Si and S2. 

Invariants. The meaning of the invariant of a single conic 
is expressed by the fact that a J = is the condition for two 
straight lines. 

Again, aV = is satisfied when the point equation of Si 
involves only product terms and the line equation S^ only squared 
terms, or when the point equation of Si involves only squared 
terms, and the line equation of S.2 only product terms. 

It is then the poristic condition f that there should be an 
infinite number of triangles inscribed in the first conic and self- 
conjugate in the second, or, what is the same thing, that there 
should be an infinite number of triangles self-conjugate to the 
first conic and inscribed in the second. We shall consider this 
type of invariant more fully in the next chapter. 

Covariants, etc. The simple ones are well known, viz. 

ttx^ = is the condition that (x) should be on the first conic, 

Ua' = is the condition that (u) should touch the first conic. 

* Reducibility itself is a purely algebraical idea and the reader will soon convince 
himself that it is generally hard to obtain any geometrical satisfaction from the 
fact that a covariant is reducible. See a curious remark of Clifford's, Collected 
Papers, p. 81. 

t The sufficiency of the condition can easily be seen by taking a triangle 
inscribed in the first conic and having two pairs of vertices conjugate with respect 
to the second for fundamental triangle. 



229-230] TERNARY FORMS 289 

Of those not involving determinantal factors, there remain 

It will be sufficient to consider one of these. Now axofaU^ =■ 
is the polar of the point {a^u^, cuug,, a^Ua) with respect to aV = 0. 

This point is the pole of u with respect to the first conic 

u^- = or a^ = 

and hence a'xd'aiia = 0, when u is constant, represents the polar 
with respect to the second conic of the pole of u with respect to 
the first. When x is constant, it is the tangential equation of 
the pole with respect to the first conic of the polar of x with 
respect to the second. 

Again {aa'uf = is the equation of the envelope of the lines 
cutting the two conies in harmonic point pairs, and in like manner 
(aa'a;)"^ is the locus of a point such that the tangents drawn to the 
two conies from it are harmonic line pairs. 

Consider now the forms involving one determinant factor. 

(i) {aa'u) a^a X = is the condition that the lines 

a^ciy = 0, a sa!y = 0, iiy = 0, 

should be concurrent, y being the current coordinate, i.e. when 
u is constant it is the locus of points whose polars intersect on the 
line u, and when x is constant it is the equation of the point of 
intersection of the polars of x with respect to the conies. 

(ii) (aa'ic) iiaiw is dual to the last ; when x is constant it 
represents the envelope of a line such that the line joining its 
poles passes through x, and when u is constant it represents the 
equation of the line joining the poles of u with respect to the two 
conies. 

(iii) (aa'u) a'aa^Ua is the Jacobian with respect to y of the 
quantities 

Equated to zero, these represent three straight lines, namely, 
the polar of x with respect to the first conic, the polar with 
respect to the second conic of the pole of u with respect to the 
first, and the line u. The vanishing of the concomitant is the 
condition that the three lines should be concurrent; hence, for 

G. & Y. 19 



290 THE ALGEBRA OF INVARIANTS [CH. XIII 

example, when u is constant, the equation represents a straight 
line constructed as follows : let Pj be the pole of u with respect to 
the first conic, and v the polar of Pj with respect to the second 
conic, then the line represented is the polar of the point {u, v) 
with respect to the first conic. 

(iv) (aa'a?) a^-a^a. is the Jacobian with respect to the i;'s of 

and hence equated to zero is the condition that these three points 
should be collinear. 

Now ujo^ = represents the pole of ii with respect to the first 
conic, and aafXaVa.' = represents the pole with respect to the second 
conic of the polar of x with respect to the first. Thus we can 
interpret the concomitant geometrically. See also Ex. ix. p. 294. 

(v) {aa'u) aa'a'aUaUa.' is a contra variant of the third class, and 
moreover the only such contravariant that can be built up from 
the members of the system. But the angular points of the 
common self-conjugate triangle must be given by equating to 
zero a contravariant of the third class. Hence the contravariant 
in question represents the vertices of the common self-conjugate 
triangle of the conies ax'^ = and aV = 0. 

We may prove this as follows * : The three conies ti^^ = 0, 
Ua'^ = 0, and {aa'uy = have a common self-conjugate triangle 
if there be a proper triangle self-conjugate to the first two, as 
we see by taking it for triangle of reference. Further, when 
three conies are referred to their common self-conjugate triangle, 
the Jacobian of their tangential equations is the tangential 
equation of the vertices. 

Hence the Jacobian of Wa^ Wa'^ {aa'uf represents the vertices 
of the common self-conjugate triangle. 

Now it is 

{aa'u) u„Ua' (aa'aa) — {aa'u) u^Uo.- (aaa'a' — aa-a'a). * 
But {aa'u) UaUa-aaa'a.' = 0, 

for otherwise it would involve the factors aa^ and aVS whereas its 
total degree is only six, so that the remaining factor would be 
a contravariant of zero degree which is impossible. 

* See also Ex. vi. p. 293. 



230-231] TERNARY FORMS 291 

Hence the Jacobian in question equated to zero is equiva- 
lent to 

{aa'u) UaUa'Cia-a'a = 0. 

(vi) By exactly similar methods we can shew that 
{aoix) aa.'Cia.a'xax = 
represents the sides of the common self-conjugate triangle. 

231. Let us now consider some problems bearing on two 
conies with a view to illustrating the second inquiry of § 229. 

(i) To find the equation of the reciprocal of the conic So with 
respect to 8^. 

If y be a point on the reciprocal conic its polar with respect to 
€1x^ = must touch Uaj'= 0. 

The polar is 

a^ay = hxhy = 0, 

and if Ux = touches u^^ = 

/ ^'^ ^2 ^2 \ '^ 

\dvidx^ dv-idxc, dv-idx^l ^ '^ °- >' 
Hence in this case 

d' d' d^ \% 



so that the reciprocal is 

axbxaa'ba' = or - (axK- - a^'hY + ax^ha,'^ + b:c-cta.''^ = 

or = la^'hoT- - {ololx)- = '±A^^. . S^ - F, 

which expresses the equation in terms of the irreducible system. 

(ii) To find the point equation of the covariant conic 

{aa'iif — 0. 

The point equation of u^ = is {a^x)- = 0, and hence in our 
case the point equation is 

[(m'hh'xY = Q, 

or {{ahh') a'^ - (a'bb') a^Y = 0, • 

i.e. (abby a'x' + {a'hh'f ax- — 2 {ahb') {abb') a^a'x = 0. 

19—2 



292 THE ALGEBRA OF INVARIANTS [CH. XIII 

Now 
(abb') (abb') a^a'x = — (abb') (a'ab') a'J)x 

= \(abb')a'x[aa'bx — ba-ax\, where a' = (a'b') 
= ^ (aoLx) (abb') a'^ 

= - i (aa'a;) (aba') b'x = -i (aa'a;) {a'^b'^ - a'^b'a] = i (aa'xf, 
thus the point equation is 

(abby aV + (a'bb'y a^ - \ (txrixf = 0, 

or J-na-'Sg + '^m'Sfi — i^=0. 

(iii) To find the locus of the point of intersection of harmonic 
pairs of tangents to F and >S. 

The locus for u^ =■ and iw^ = is (aaxf = and in our case 

the locus is accordingly 

[aa'axY = 

or (aa;a'a — a'jgaa)^ = 0, i.e. /Sfi^iia + 'Sa-^m — 2aa;a'a;aaa'a = 0. 

To reduce the last term we perceive that it must contain J-m 
and we write it 

(abc) a'x (a'bc) «« = i (abc) a'x [(a'bc) ax + (caa') bx — (aa'b) Cx] 
= ^ (abc) a'x (bca) a'x = i A^i . S^. 
The equation required is 

^1-^112 + ^5^2-^ 111 ~ f 'Sia^iii = 0, 

or Aiu82 + SSiAu2 = 0. 

The following gives an easy means of verifying the results of 
(i), (ii), (iii) above and of the examples which follow. 

Taking the conies in the canonical forms 

Si =zaiXi^ + a^xi + a^xi 

O2 =^ X-i + x<^ + x^ 

we have 

Xi = 2 (a^a^vi^ + a^a-iui + a^a^u^) 

S2 = 2 (Ui^ + «2^ + ih% 

^iu = 6a,a2a3, ^222 = 6, ^uj = 2 (ajas + as^i + Oiaj), 

^122 = 2 (oi + aa + ftj), 

^ = (ttj + rts) «i^ + (^3 + aO ui + (ai + Oa) 2*3^ 

jP = 4 {tti (Oa + ag) «i'* + as (03 + Oj) aja^ + 03 (a^ + aj) ajg^j . 



231] 



TERNARY FORMS 



293 



where 



Q = 



Ex. (i). The locus of points whose polars with respect to S2 cut S^, S.2 in 
pairs of points harmonically conjugate is 

{aa'b') {aa'c') b'xc'x = 0, 

or A.222^i-SAi22>%=0. 

Ex. (ii). Prove that 

{abuy=^Q^ax^byhi,^) 

_s_ _3_ _a_ 

dxi dx2 8^3 

d_ d_ d_ 

^yi 8^2 ^Vz 

AAA 

92, 3^2 823 
and hence that (a^, a'/i', uY=^{<-A^^^A2a^{aa'iCf. 

Interpret this result geometrically. 
Ex. (iii). The line equation of F=0 is 

(aa! ^ m)2=0, 

or {a^^f uj- + (a'^^')' «.' " H^ ^' «)'• 

Thence 

(^' ^' w)2 = A{3^iii.4i2.^22+ 3^222-^11221 -2.4 111^222*}- 

Cf. Ex. (ii). 

Ex. (iv). Deduce from Ex. (iii) that the discriminant of F is 

{ad ^^ ryy = il (9^112^122 - -^111^222) ^111 -^222- 

Ex. (v). Deduce from Ex. (ii), 

(aa' 66' cc')^ =^(9^ j 12 ^122 --4 111^222) • 
Ex. (vi). Prove that the discriminant of {aa'u) a^a'^ is 

(aa'u) {bb'u) (cc'u) {ab'c') (a'bc). 
Hence if the conic (aa'u) axa'x = be two straight lines 
(aa'u) a ,a' u u ,—0. 

\ ' a a a a 

Thence verify that this equation represents the angular points of the 
common self-conjugate triangle and work out the dual results. 

Ex. (vii). Prove that if the point equation of a conic be 

Xi>S'i + X2'S'2 = 0, 
then its tangential equation is 

and its discriminant is 

Xi3J„j + 3Xi2X2.-lji2 + 3XiX22J,22 + V^222- 

Ex. (viii). Prove that the point equation of any covariant conic is of 
the form 

Xi,s'i+X2.y2+xi^=o, 



294 THE ALGEBRA OF INVARIANTS [CH. XIII 

where Xj, X.,, X are invariants, and that its line equation is 

Xi22i + 2XiX2* + X2^22 + §XXi(Jiil22 + 3Jl222i)+|XX2(^2222i + 3Ju222) 

+ X2A(3^jj^^j2222 + 3^222-^n22i-2Jin^222*) = 0- 

Ex. (ix). The locus of points whose polars with respect to S^ and F meet 
on the line u is 

(a aa u)ax{aa'x) = 0, 

or {aax)a^;u^a^ = 0. 

This gives a simpler interpretation of the irreducible form than that 
given in § 230. 

Ex. (x). Use the method of Ex. (ix) to interpret 

(aa'u)a'au^=0. 

^ * a. X a. 

Examples (ix) and (x) enable us to interpret all the irreducible mixed con- 
comitants very simply in connection with the conies S^, /S'g, F, *. 

Ex. (xi). The locus of points whose polars with respect to F and * meet 
on u is 

^112 {aax) a^u^a^, + ^122 {aa'x) a'^u^, a^,=0. 

(Use the point equation of * given in Ex. (ii).) 

Ex. (xii). The equation of the line joining the jjoles of u with respect to 
F and * is 

^222-^112 (««'«) '^a%^'a + "^m ^^ 122 (««'«) «a'<*'a;«a' = ^^ 

(Use the result of Ex. (iii).) 

Ex. (xiii). Calculate the four invariants of the conies S^ and F. 

We have 

^222 = 2^ (9^112^122 -^111^4222)^111^222 by Ex. (iv). 

(7,22 = (« ^' 0/3T = M3^'liii ^2^22 + ^222^112^111)- 

^112 ==3 -^111 -^122 • 
^111 = -^111- 

232. Gordan's general method. Consider a concomitant of any number 
of forms containing the r letters a, b, c, ... h, L If we replace each factor of 
the type (aku) by a^, each factor of the type (abk) by {abu) and delete all 
factors kx the resulting expression is still an invariant form but only of degree 
r — 1 because it only contains r—1 symbols. 

Hence reversing the operation, i.e. replacing a proper number of factors 
of the type a^ by (aku), some of the type (abu) by (abk) and introducing a 
sufficient power of k^ we can deduce the form of degree r from one of degree 
r-1. Applying this process in all possible ways to all invariant forms of 
degree r - 1 we certainly obtain all invariant forms of degree r. We pause to 
explain more precisely what we mean by applying the reverse process in all 
possible ways to an invariant cf). Suppose in fact that the newly introduced 
letter k belongs to a form of order n, then we replace any p factors of the 



231-233] TERNARY FORMS 295 

type ax by iaku) any q factors of the type {ahiC) by {ahh) and multiply by k^ 
where of course we must have 

and we must take all values of jo, q, r satisfying this equation subject, of 
course, to the condition that there exist p factors of the given type to alter 
and a like restriction for q. 

For example, let the original form be 

{ahu){acu)hxCx, 

and suppose the new letter d like a, b, c belongs to a form of order 2. 

We can only change 2 factors at most in this case and we have 

p+q — l or 2, 
there are five cases, 

p = l, q = 0; p = 0, q=l', p = 2, q=0; p = l, q = l; p = 0, j = 2; 

and in following out the case 2) = l, q = l, r = 0, for example, we deduce four 
forms from the given one, viz. 

(abd) {acu) (bdu) c^ , 

{abd) {acu) b^ (cdu), 

(abu) (acd) {bdu) c^, 

{abu) {acd) b^ {cdu). 

The above indicates the general method of prcjcedure, but some introductory 
lemmas are necessary to render the method of any practical value — for 
example for all we know at present a form of degree r—\ which is identically 
zero might lead to irreducible forms of degree r, and we need hardly say that 
this would complicate matters enormously. 

233. The reader will have observed some likeness between the above and 
the methods used in Chapter v. on binary forms to deduce the invariants of 
degree r from those of degree r- 1. This analogy will be fiurther exemplified 
in the rest of the argument. 

Consider the eflfect o^ the operator ii^' on the expression 
<f) being a covariant of degree r - 1 and 12 being the operator 



d 


d 


d 


3a^i 


3^2 


dxs 


3 


a 


d 


9^1 


^y'i 


0^3 


d 


d 


a 


dz^ 


dz. 


dz^ 



As we saw in the last chapter, the result is the sum of a number of terms 
each containing the determinantal factors of together with p of the form 
{akic), there being in the end p fewer factors of the type a^, {n-p) factors X-^ 
and no factors of the type u^ in each term. 



296 THE ALGEBRA OF INVARIANTS [CH. XIII 

Next operate on the complete result of which a typical term is ■\^ky'^~P 

d^ 32 52 

q times in succession. 

Since the effect of this operator on {abu)ky^~p is to give a multiple of 
(ahk)ky^~P~\ it is very easj' to see that the effect of the operator q times' 
is to give a number of terms each containing the same determinantal factors 
as ■<\r, with the exception that q factoi-s of the type {ahu) are replaced by {abk) ; 
the power of ky remaining in each term is ky^'^'t, and k replaces « in q 
places in all possible ways. 

Hence if we operate with QiQP on the product (f>ky"Ui'P and then put )/ = .v 
we obtain the sum of a number of terms each of which has the same deter- 
minantal factors as except that in any q of them u is replaced by k and p 
new ones of the type (aku) are introduced while p factors of the type a^ 
disappear and finally the factor ^^""P-* is introduced in each term. 

Consequently in the resulting expression there will be contained every 
term derived from <ji in the reverse process explained above with these 
definite values of p and q. 

We may conveniently call 

the transvectant of (f> and k,.^ whose indices are p, q, the order of the indices 
being essential, and we have the resvdt that every concomitant of degree r is 
the sum of a number of terms each occurring as a term in a transvectant of 
a form of degree r — 1 with k^": Naturally when there are different forms we 
have to introduce in turn a symbol belonging to each. 

234. We next require certain relations that exist among the terms of 
the same transvectant, and to establish them we shall alter our notation for a 
moment. 

Suppose in fact that 

= aj^) aj^) . . . ajp') u^ u^^ ... u^ ,M, 

where M contains neither u nor x: 
In each term of the transvectant 

we have q of the u's replaced by k, p of the terms a J'') by (aku). 

We shall call two terms JV^ and aVg adjacent when jo + y — 1 of the factors 
affected in (f> to obtain them are common, and two cases will arise according 
as the remaining factor affected is an a^ or a m . 

In the first case we have, supposing that aj^) and aj*) are the additional 
altered factors in the two terms respectively, 

iV^i - A\ = N {(a('-)/{-M) aj^') - («W/-m) aj^'')] 

=N{{ai>-)ai')k) u^ - (aWa(»)M) k^}. 



233-234] TERNARY FORMS 297 

Now iV(a<'")a(*)ifc)is a term in 

where ^^ is deduced from by changing aj^'^)aj^) into (at'") «(*)«) and further 

ir(a('-)a(»)tt)ifc^ 

is a term in {({y.^, kx"}"'^'^ 

where <f>2 is the same as (f)^. 

In the second case let ic^ and it^ be the additional factoi-s altered in the 
• J 

two terms, then 

= —^^'{aiOjhl} 

and this latter is a term in 

where ^3 is deduced from by changing u u^ into {oiajx). 

Now if we call the sum of the order and class of a function its grade it is 
evident that (f>i, (f>2, (f)^ are each of grade less by unity than that of 0. 

Further between any two terms of the transvectant we can insert a 
number of others such that any two of the whole sequence are adjacent in 
our sense of the word and accordingly we have the important theorem : 

" The difference between any two terms of the same transvectant can he 
expressed in terms of transvectants of functions of loicer grade than (f) 

with i'^\" 

Thus if we consider our function <p of degree r — 1 in ascending order 
of grade we need only retain one term out of each transvectant that we 
consider — or if we please the sum of any number of terms will equally serve 
our piu^)ose and in particular the transvectant itself might be used. It 
follows at once that if a transvectant contains a single reducible term it may 
be neglected entirely. 

Again, if there be a linear relation among a number of the forms of 
degree r — l there will be a linear relation among the transvectants of given 
index formed from them, so that we need only consider linearly independent 
forms of degree r — l. In particular, zero forms of degree r—l can be entirely 
put out of account. 

A knowledge of the irreducible system up to and including degree r — l 
therefore gives us immediately all the forms ({> of which transvectants need 
be considered, for we have only to include the irreducible forms of degree r — l 
and such simple products of the others as are of total degree r- 1. 

We have nosv eflfecte^l our purpose of making the method at present under 
discussion of real value, and we proceed to illustrate it by reference to the 
complete system for a single quadratic. 



298 THE ALGEBRA OF INVARIANTS [CH. XIII 

235. Quadratics. We have here five different seta of indices for trans- 
vectants, namely 

^ = 1,^ = 0; ^ = 0,2 = 1; ^=2,y = 0; p = l,q = \; p = 0,q^2. 

Consider now how far products need be taken into account, if p + q = l 
then all products may be neglected because only one factor is modified 
and hence some terms of the transvectant of a product are certainly re- 
ducible. If jD = 2, g' = 0, then a transvectant of ^j^o with k^^ will contain 
reducible terms unless the orders of ^^ and f^, ^^^ both unity, also for 
^=0, ^=2, we need only consider in like manner products of two forms whose 
class is unity. If p = l, j = 1, we need only consider the jiroduct of two forms 
when one is of zero order and the other of zero class. Throughout products 
of more than two forms need not be taken into account. 

Further, in every case pure invariants of degree ;■— 1 can give rise to no 
new forms. 

236. Single Quadratic Form. Of the first degree we have 

Ctx' = Ox = Cx = . . . 

Proceeding to the second degree we have 

{abu) ttxhx for p = \, q = 
and {abuY for p = 2, q=0 

of which the first is zero. 

Third degree. From {abuY we get 

(abc) (abu) Cx (01) 
(abcf (02) 
and from Ox^ftx^ we can only get reducible forms. 

Now {abc) {{abu) Cx) = \ iabc) {(abu) Cx + (bcu) a^ + (abc) iCx} 

= l{abcfux' 
so that of the third degree we have only the invariant {abcf. 

Fourth degree. From {ahafc^ we need only consider 

{abd) (abu) (cdu) Cx 
arising from j9 = 1 and 5' = 1. 

This is J (abu) (cdu) {{abd) Cx + {bdc) ax + {dca) bx + {cab) dx\ = 0. 

For further forms we need only consider transvectants of products of 
powers of Cx^ and {ahif with p+q:!^2. 

These all contain reducible terms and hence there are no new forms 
so that the complete system consists of Ux^, {abiif and {abcf as ah-eady 
indicated. 

For the case of three quadratics and incidentally two see Baker, Camb. 
Phil. Trans. Vol. xv. 



CHAPTER XIY. 

APOLARITY {continued). 

237. Apolar Conies. Two conies 8 and S' whose equations 
in point and line coordinates are respectively 

S = ax = axx + 63^2- + cxi + ^fx^x-i 4- 2gx.iXi + Ikx^x.^ = 0, 

and 2' = uj = ^ V + ^V + ^''"s' + 2i^'?t2«3 + 2G'usU, + 2ir'wiW2 = 

are said to be apolar when the invariant tv", or what is the same 
thing, 

aA' + hB' + cC + %fF' + 2gG' + 2hH' 
vanishes. 

This relation between the two conies is not a symmetrical one, 
inasmuch as it arises from the point equation of one and the line 
equation of the other; it is convenient to have an alternative 
name shewing the exact relation between the curves. For 
reasons to be explained later we shall say that S is harmonically 
inscribed in S', and that S' is harmonically circumscribed to S. 

The curves are also apolar when a'^- = 0, but in this case S' is 
harmonically inscribed in S. 

As is well known from the geometry of conies, a^'- = is the 
condition that there should exist an infinite number of triangles 
self-conjugate to S and circumscribed to S', or an infinite number 
of triangles inscribed in S and self-conjugate to S' — in fact the 
equations a^' — and ?/„'- = can be so transformed that the first 
has no product terms and the second has no square terms, or that 
the first has no square terms and the second has no product terms. 

The relation 0^ - = is linear in the coefficients of the equations 
and Ua'- = 0, 



300 THE ALGEBRA OF INVARIANTS [CH. XIV 

hence a conic apolar to the conies Si, S.2, ... Sris apolar to any conic 

XiOi + X2'^2 + • • • + \r^r ^^ 0. 

Further if these r conies are linearly independent there are 
(6 — r) linearly independent conies apolar to them. In particular 
there is a unique conic apolar to (harmonically circumscribed to) 
five given linearly independent conies. 

The same remarks apply to p given conies 

2/ = o, 2; = o,...v = o. 

and in particular there is a unique conic apolar to (harmonically 
inscribed in) five given conies. 

238. Particular Cases. (i) If a^^ = represents two 
straight lines and Ua.^ = 0, the two lines are conjugate with respect 
to Ua^ = 0. If a^ = represents two straight lines coinciding in I 
then the line I touches the conic u^'- = 0. 

(ii) If Ua'^ = represents two points then these points are 
conjugate with respect to a^^ = 0. If Wa" = represents two points 
coinciding in I then the point I lies on the conic a^ = 0. 

All these statements can be verified immediately by using 
the apolar condition expressed in terms of actual coefficients or 
symbolically, e.g. if aj^ = v^Wx the apolar condition is 

which is the condition that the lines Vx = 0, Wx = should be 
conjugate with respect to iia'"^ = 0. 

239. Ex. (i). If two pairs of opposite vertices of a complete quadrilateral 
are conjugate with respect to a given conic so also is the third pair. 

Let the conic be a^^ — o, and suppose the two pairs of opposite vertices are 
given tangentially by 

The general equation of a conic inscribed in the quadrilateral is then 

\UpUp'+HUgUg'=0, 

and since UpUp' = and UgU^=0 are both apolar to ax^=0 it follows that every 
conic inscribed in the quadrilateral is apolar to ax^ = 0. But the third pair 
of opposite vertices is one such conic, hence these remaining vertices are 
conjugate with respect to the given conic. We shall call such a quadrilateral 
a quadrilateral harmonically inscribed in the conic 0x^ = 0. 



237-240] APOLARITY 301 

Ex. (ii). Four conies have in general one common harmonic quadri- 
lateral. 

Let the conies be S^, S^, S^, S^, then the apolar system is of the type 

consequently the apolar conies in general all touch four fixed straight lines. 
The opposite vertices of the quadrilateral formed by these lines taken in pairs 
constitute conies of the apolar system, and hence pairs of opposite vertices are 
conjugate with respect to each of our four conies. Hence the quadrilateral is 
harmonically inscribed in each of the given conies. 

Ex! (iii). A triangle ABC and its polar triangle with respect to a conic 
are in perspective. For if the polars of B and C meet the sides CA and AB 
respectively in Q, R, and the line QR meet BC in P, then the quadrilateral 
formed by BC, CA, AB and the line PQR has two pairs of opposite vertices, 
viz. {B, Q), (C, R) conjugate with respect to the conic ; therefore {A, P) are 
conjugate with respect to the conic, or the polar of ^ meets BC m. P. Thus 
the polars oi A, B, C meet the opposite sides in three collinear points, and 
they therefore form a triangle in perspective with ABC. 

Ex. (iv). If u^W = 0, «/^) = 0, Mx<^) = 0, Wx(*) = 

be the sides of a quadrilateral harmonic with respect to the conic S=0, then 
we have 

S=Xiui\^+\u(%^ + \su(%^ + XiU(%^. 

For let two pairs of opposite vertices be (pp') and (qq') ; then apolar to the 
conies (pp) and (qq') we have the five conies 

v.(i):,2 = 0, ^(2)^2=0, id%^ = 0, u(%^ = and S=0. 

But the first four are linearly independent and hence S is a linear combination 
of them. 

240. Some interesting applications can also be made to the 
metrical geometry of conies. 

In fact, suppose that the tangential equation of the circular 
points at infinity (I, J) is 

<f) = tiy- = 0. 

Then a conic apolar to (f> has /, J for conjugate points and is 
therefore a rectangular hyperbola. 

Again, the tangential equation of a circle whose centre is j9 is 

of the form 

Up^ = \<^ 

where \ varies with the radius. 



302 THE ALGEBRA OF INVARIANTS [CH. XIV 

If a circle G be apolar to a conic 

S = w,^ = 

then the director circle of the conic cuts the circle C orthogonally. 

Use rectangular Cartesian coordinates, and let the equations 
of the conic and circle respectively be 

AP + 2Elm + Bill" + 2GI + 2Fm +0 = 0, 

and a^^ + 2/2 + 2gx + 2/3/ + c = 0, 

so that we have 

A+B + 2gG + 2fF+cG==Q. 

But the equation of the director circle of the conic is 

G{x' + y'')-'lGx-^Fy + A-\-B = 0, 

and this cuts the given circle at right angles if 

^ G ^.F A + B _ 

le.ii 2gG + 2fF+cG + A+B = 

which is precisely the condition of apolarity. 

The director circle of a conic inscribed in a triangle cuts the 
self-conjugate circle oHhogonally. 

For since the self-conjugate circle has the triangle for a 
self-conjugate triangle and the conies are inscribed in the triangle, 
each of the conies is apolar to the circle. Hence their director 
circles cut the self-polar circle at right angles. Or thus, — the 
system apolar to the inscribed conies is of the form 

M^x + mx^ + vi'^^ = 0, 

where p^ = 0, qx = 0, rx = represent the sides of the triangle. 

By suitably choosing \, fj,, v this equation may be made to 

represent a circle, and from the form of its equation it is the 
self-polar cu-cle of the triangle. 

The locus of the centre of a circle which has two fixed pairs of 
conjugate lines is a rectangular hyperbola. 

In fact, suppose the lines are 

px = 0, qx = O1 
rx = 0, Sx = OJ 



240-241] APOLARITY 303 

The system apolar to the tangential system of conies having 
these two pairs of conjugate lines is 

^PxS'x + f^r^S:, = 0. 

There is one value of the ratio X : /* for which this represents 
a rectangular hyperbola. 

Let >Sf = be the rectangular hyperbola in question and let 

U,f = \^ 
be one of the circles. 

Then B is apolar to ^, because <S = is a rectangular hyperbola, 
and as it is apolar to 

Up^ = \(f) 

it is apolar to i/,/. Hence the point jj must lie on S and therefore 
S is the centre locus of the circles. 

In general when a rectangular hyperbola S is apolar to a 
circle 2 the centre of the circle lies on the rectangular hyperbola. 

241. Apolar Curves in general. The two curves whose 
equations are 

and </> = Ua^ = OJ 

are said to be apolar when the form a^ar^^~'^, which we denote by 
•v|r, is identically zero. 

Except for a numerical multiple we have 

' \dUxOX^ 011^0X2 ou.oxj 

The following are analogous to theorems on binary forms. 

I. The form <j) is apolar to any polar of f whose order is not 
less than n. 

For {a^'''~''ay'-, iC}"''' = a^a^-^-'^ay"" 

and this is zero as we see by polarizing the identity 

«»''«*"'"'' = 
r times with respect to y. 



304 THE ALGEBRA OF INVARIANTS [CH. XIV 

II. The fovTn f is apolar to any form which contains (f) as a 
factor and whose class does not exceed m. 

For if <^' be any form of class n we have 

{/, ^,^'|o.»+.' = |(y; ^)o.n,^'|o..' 

= 0, 

since (/, </>)"''' vanishes identically. Hence / is apolar to 0<^'. 
We have supposed that m'^n hitherto. Exactly similar remarks 
apply to the case in which n > m. 

The search for the forms of given class (n) apolar to a given 
form / is facilitated by the fact that the necessary and sufficient 
conditions for <f) are that it should be apolar to every (m — w)th 
polar of/. 

For an (m — n)th polar is 

and this is apolar to Wa" if 

aa^ay"'-'' = 0. 

But if this relation be true for all values of y then the form ^ is 
apolar to/. 

242. Ex. (i). A ternary cubic has three linearly independent apolar conies. 
For the first polars of the cubic are linear combinations of 

dx-^ ' dx^ ' dx^ 

which are three linearly independent quadratic forms. Hence there are three 
linearly independent conies apolar to all first polars and therefore apolar to 
the cubic itself. 

Ex. (ii). A ternary quartie has an apolar conic mdy when the deter- 
minant of the coefficients of its second differential coefficients vanishes. 

For an apolar conic must be apolar to all second polars and they are linear 
combinations of 

82/ a^ a^ _^f ay ^f 

dx^^ dx^^' 8-^3^' 8.^23.^3' dx^dxi' dxidx.2' 

In general there is no conic apolar to each of these six, but there will be 
an apolar conic if the six be not liuearly independent, i.e. if the determinant 
of six rows and six columns be zero. 

243. G-eneral Theory of Curves which possess an 
Apolar Conic. By using suitable coordinates the analysis of 
ternary forms apolar to a given conic may be reduced to that of 
binary forms. 



241-243] APOLARITY 305 

Suppose that the fixed conic is 

x-^x^ — xi = 

or in line coordinates 4iUiU^ — u^ = 0. 

There is no loss of generality in taking the equations in this 
form, because by suitably choosing the triangle of reference, the 
equation of a proper conic can be always reduced to the form 

13 ^~ '* — 

Thus we may take for the parametric representation of points 
on the conic 

X^ = Vi, X2 = I'l ^'2 > ^3^^ ^2"> 

and we shall call this point (^1, x^, x^) the poiut v. 

If the line u^^^O meet the conic in the points (\ yx) the 
quantities X, /m are given by 

UiVi' + U2V1V2 + u^v^ = 
so that Ui = X.2/A2 

«2 = - (^1/^2 + ^2/^1) ■ (A> 

Us = Xi/ii 
except for a constant factor. 

Hence we may regard the quantities 

as the coordinates of the line, and a homogeneous relation of 
order m connecting the u's becomes a homogeneous symmetrical 
relation of order 2m between the X's and /x's. Thus any 
symmetrical relation between X and fi is equivalent to the 
tangential equation of a certain curve whose class is one-half of 
the order of the given relation. 

The coordinates of the tangent at the point v are 

and hence the points of contact of the tangents from the point x 
to the curve are given by 

x^vi — Ix^V-^Vo + x^v^ — 0, 

so that we may take 

X\ = Xi/^i j 

Xs = Xa/ia j 

G. & Y. 20 



306 THE ALGEBRA OF INVARIANTS 

[CH. XIV 

Consequently a homogeneous symmetrical relatio. 

the \'s and /i's of order 2n represents a curve of order n. '^' ^ . 

given a symmetrical relation we therefore deduce two cu ° 

equations from it, one in line coordinates and the other in poii ' 

coordinates. The curves represented are reciprocal with respecb 

to the fundamental conic because, taking X,, /j, to be fixed, the line 

u given by (A) is the line joining them, and the point a; given 

by (B) is the intersection of the tangents at A, and /i, i.e. the pole 

of the line u. 

This method of representing a point by the parameters of the 
tangents drawn from it to a fixed conic and a line by the 
parameters of the points in which it meets the conic was 
practically used by Hesse and first explicitly used by Darboux. 

244. By means of this system of coordinates we can readily 
find all curves apolar to the given conic. 

I. Suppose that w^" = 

is a class curve apolar to the conic, then we have 

(72' - 7173) V = 0. 
Thus (72^ — 7173) multiplied by any function of the 7's is zero 
if it be interpretable, hence this symbolical expression must be 
zero, and we may write 

7l = «2', 

72=-aia2, 

73 = «lS 

and our equation is 

The a's are now the only symbols used, and it is clear that 
as any expression of degree n in the 7's represents an actual 
quantity, any expression of degree 2n in the as is an actual 
coefficient, or in other words the as are the symbols of a binary 
form of order 2n. 

On introducing the Vs and fi's our equation becomes 

[a^%ifi2 + Oitta (\i/*2 + ^2/*i) + ai^A-i/ii}** = 0, 

or {(OiXi + ttaXj) (tti/Ai + a^fi^Y = ^' 

that is finally ax^a/ = ^- 



243-244] APOLARITY 307 

The binary 2n-ic of wliich the as are symbols has an 
important significance, for if we make X. = //., the line Ux is the 
tangent to the fundamental conic at the point X; consequently 
the equation 

a^' = 

gives the parameters of the points of contact of the 2n common 
tangents of the apolar curve and the conic. 

Conversely, when the equation 

ax^" = 

is given, the equation aA."a/' = 

is uniquely determined by polarizing, and hence we have the 
theorem that a class curve apolar to a conic is uniquely deter- 
mined when its common tangents with the conic are given. 

By proving this theorem from first principles, and then 
observing that 

a^^'a^'' = 0, 
or its equivalent 

{a^'Ui — a-ia^iio, + o.^u^''^ = 0, 

certainly represents a curve apolar to a conic, we can shew that 
any apolar curve may be*reduced to the form 

ttA^a^^' = 0, 

without using a parametric representation of the symbolical 
equation 

II. Suppose that the curve 

c/' = 

of order n is apolar to the given conic 

4wiW3 — u^ = 0, 
then we must have 

(4ciCs-c,»)c^"-^=0, 

and reasoning as before, we have 

* See Schlesinger, Math. Ann. Band xxii. 

20—2 



308 THE ALGEBRA OF INVARIANTS [CH. XIV 

We may now use the parametric representation 

Ci = ttiS 

C3 = fl2 , 

and the equation of the curve becomes 

(ai^nci + la-fl.^^ + a^^x^Y = 0, 

or introducing the X's and /a's 

{a^Xifii + a-^a^ (Xi/Ua + Xojx^) + a-fXif^o]'^ = 0, 

i.e. a^'^a^'' = 0, 

and as before the a's are the symbols of the binary 2n-ic a^''", 
which, equated to zero, gives the points of intersection of the 
apolar curve and the conic. 

Example. To find the conic apolar to x^x^ — x.^ = Q ivMch touches the 
tangents to this conic at the points given hy v^^-v.^ = 0. 

Here a\2n = is Xi*-X2* = 0, 

thence ax"a/u,"=0 



("B-xT'V-V). 



or XiVi'^-X2V2^=0. 

On using the substitutions 

the equation becomes u^ — u^ = 0, 

or («3+Mi)(W3-«l)=0. 

so that the conic consists of the two points (1, 0, 1) (1, 0, — 1), and in fact it 
is easy to see that these points are conjugate with respect to the conic. 

245. Theorems on conies apolar to the fundamental 
conic. The equation of a conic apolar to x^x^ — x-^^Q, and 
touching the tangents at the points given by 

a/ = 0, 

is equivalent to a^^a^^O, 

and hence to m/ = 0, 

where Uy = a^a^. 

Now suppose that A, B, C, D are four points (X, fj,, v, p) on 



244-245] APOLARITY 309 

the fundamental conic, and that the lines AB, CD whose equa- 
tions are 

Vx = and w^ = 0, 

are conjugate with respect to the apolar conic Uy^ = 0.' 

Then since the condition of conj ugacy is VyWy = and 

Vy = iif^a^ , Wy = a^a^, 

we have 

a^^a^ap — 0, 

that is the quartic giving \, fi, v, p is apolar to a/ = 0, 

This is one of the simplest geometrical representations of forms 
apolar to a given form. From the symmetry of the result, we 
see that each pair of opposite sides of the quadrangle ABGD are 
conjugate with respect to the apolar conic. 

Thus there is an infinite number of quadrangles inscribed in 
the fundamental conic and harmonic to the apolar conic ; the four 
vertices are given by fo^ms apolar to the form 

a,' = 0. 

Now if X, yu-, V be chosen so that p is arbitrary, any line 
through A is conjugate to EC, so that A is the pole of BG, and 
hence ABC is a self-conjugate triangle of the apolar conic m/ = 0. 
In this case the cubic giving \, /x, v is apolar to a^^ = 0, and therefore 
to every first polar of this form ; hence there is an infinite number 
of triangles inscribed in the fundamental conic and self-conjugate 
with respect to the apolar conic, and their vertices are given by 
the singly infinite number of cubic forms apolar to a^* = 0. 

Next suppose that the linear factors of the quartic giving 
A,, /A, V, p are Ij, m^, iij, i\, then 

where X, M, K, R are independent of r. 
By polarizing we obtain the identity 

a^a,^ =-- L {kl,y -H M {vnm.y + N{n^n,y + R {r^r^)\ 
where f, r^ are any two points on the conic. 
Now by means of the usual substitutions 
"l = 1^2-^2, 

U^ = - (^1^2 + ^2'7l), 
"3 = 11%, 

the left-hand side becomes Wyl 



310 THE ALGEBRA OF INVARIANTS [CH. XIV 

Consider next the term l^l^. 
Since ^1 = X2 ^■iid Z2 = — Xi, this becomes 

or Xi^Ui + \1X2W2 + Xa^Ws , 

and Xj^, X1X2, Xg^ are the coordinates of ^, so that 

hh — ''■''A > 
where w^ = is the tangential equation of A. 
Hence we have 

m/ = Xii/ + Mub^ + Nuc" + Rub'', 
and the conic is represented as the sum of four squares. 

In particular, ii A, B, G he the vertices of a self-conjugate 
triangle, we obtain in like manner 

V = Lua^ + Mub^ + Nu(?. 
These results are well known and easily obtained otherwise, 
* but the methods here used may be applied with equal success to 
more difficult problems as we shall presently shew. 

Exactly the same reasoning applies to a conic c^ apolar to 

4iVriUz — U2 = 0, 

and now the triangles are circumscribed to the fundamental conic 
and self-conjugate to the apolar conic. 

246. Condition of apolarity of two conies apolar to the 
standard conic. 

Suppose the two conies are 

Cy? = 0, Wy- = 0, 

that the first meets the standard conic in the points a^* = 0, and 
the second touches the tangents to the standard conic in the 
points 6a^ = 0. 

Then we have 

Cx = axa^ and Uy = bj)^, 
and in particular 

C2 = 2a^ar, [ , 72 = - 6162 [ , 
C3 = ai ) 73 = 61^ i 

hence c^ = a^h^ + a^h^^ - 2a^a.hJ)2 = {ahf, 



245-247] APOLARITY 311 

and the conies are apolar when Cy = 0, that is when 

(aby = 0, 

or when the two binary quartics a/,* and 6x* are apolar. This gives 
another simple geometrical representation of apolar quartics. 

247. To many of the theorems developed for conies there are 
analogues for all curves possessing an apolar conic. For brevity we 
introduce a definition. 

If the equation of a curve of order m be 

J {Xi, X.2, 0C2,) = U 

and/ can be written as a linear combination of the forms 

u^wm^ r = 1, 2, ... w, 

then the n lines Ma;*'"' = 

are said to form a conjugate n-line with respect to the curve. 

In like manner if the tangential equation of a curve of class 
m be 

and ^ can be written as a linear combination of the forms 

w^J^ r=l,^,...n, 
then the n points Ux =0 

are said to form a conjugate n-point with respect to the curve. 
Suppose then that the curve 

is apolar to the fundamental conic 

1 *^3 """ 2 — ^> 

the equation may be written 

where ax^" = 

gives the 2n points in which the curve meets the conic. 
If aK^^ be apolar to the form 

whose linear factors are 

Pk''\ Pk'\ ... Pk^'K 



312 THE ALGEBRA OF INVARIANTS [CH. XIV 

then a\^ can be expressed as a linear combination of 
so that af'a^^ is a linear combination of 

and hence just as in the case of the conic it follows that the 
tangents to the conic at the points given by 

i.e. by 6/ = 0, 

form a conjugate r-line with respect to the curve whose equation 
can accordingly be expressed as a sum of r ??th powers. 

The above will suffice to indicate the general principles which 
we shall now apply to the ternary cubic and quartic. 

248. Ternary Cubic. A cubic curve, as we have seen, 
§ 242, always possesses an infinite number of apolar conies. Take 
the fundamental conic for one of these and let 

c^' = 
be the equation of the cubic. 

This may be written 

and meets the conic in the points given by 

ax« = 0. 

This binary sextic has three linearly independent second 
polars, and therefore a singly infinite number of apolar binary 
quartics but not in general an apolar cubic. 

Hence a ternary cubic may be written in an infinite number 
of ways as the sum of four cubes for each apolar conic it possesses, 
but not in general as the sum of three cubes for an arbitrary 
apolar conic. 

The condition that the binary sextic 

may have an apolar cubic is that the determinant formed by the 
coefficients of its third differential coefficients may be zero, i.e. that 
any four third polars may be linearly dependent. 



247-249] APOLARITY 313 

This condition is 

(aay {a'a'J {a"af {aa'y {a a" J {a" a'")- = 0. (§ 187) 
Now ai^ = Ci, 

2a^a2 = c.2, 

hence {aa) {a' a") {a" a) 



= Hcc'0 



i a'? a/'a," a\ , 
and the condition reduces to 

(cc'c") (c'c"c'") {c"c"c) {c"'cc') = 0. 
This is an invariant of the cubic 

and hence however we choose the apolar conic we cannot reduce 
the cubic to the sum of three cubes unless a certain invariant of 
degree four vanishes. 

249. Ternary Quartic. Here there is no apolar conic 
unless the six second differential coefficients are linearly de- 
pendent, i.e. unless a certain invariant called the Catalecticant 
vanishes. 

Now if the quartic can be written as the sum of five fourth 
powers it must have an apolar conic, because a conic can be chosen 
apolar to any five fourth powers — in fact we have only to describe 
a conic touching the five straight lines represented by the linear 
forms. 

Hence in general a ternary quartic cannot be expressed as the 
sum of five fourth powers. 

But if the catalecticant be zero there is an apolar conic and, 
taking it for a fundamental conic, the equation of the quartic may 
be written 

where ai^ = gives the points of intersection with the conic. 

Now a singly infinite number of quintics can be found apolar 
to a binary octavic, hence in this case the quartic curve has a 



314 THE ALGEBRA OF INVARIANTS [CH. XIV 

singly infinite number of conjugate five-lines, and all such lines 
touch the apolar conic. 

250. We shall conclude this chapter with a brief account of 
the class of invariant forms known as combinants, confining 
ourselves to binary forms. 

An invariant or covariant of any number of binary forms 

/l. Jly ••• Jr 

of the same order is said to be a combinant if it be unaltered, 
except as regards a factor independent of the forms, when each 
form / is replaced by a linear combination of the type 

lifi + kf2 + •'■ +lrfr, 

in which the I's are constants. 

For example in the case of two binary forms we have 

(hfi + kA, m^fi + m^A) 

= {lm){fj,\ 

so that the Jacobian of two binary forms is a combinant. 

For the sake of brevity we shall deal with the combinants of 
three binary forms 

/a = &o^i" + n&i«i"~'«o + . . . + hnX^"" 

J 3 ^ Cq Xi "T nCi Xi X2 + . . . "T Cji x^ . 

A combinant is not only unaltered by a linear substitution 
efiFected on the variables, but also by a linear substitution of the 
type 

a/ = liar + rriibr + n^Cr 

br = hdr + m.ibr + Tl^Cr 
Cr = ^ttr + Wfs^r + ^hCr 

effected on the coefficients. 

Regarded as a function of the coefficients the combinant is 
therefore an invariant of the linear forms 

dr^i + K^i + Cr^S, r=l, 2 n, 

because if we put 

^i = li^i +^2 +h^3' etc. 



249-250] APOLARITY '^^^F 315 

we find a/ = lia,. + niibr + n^Cr etc. 

which are the substitutions above. 

Hence, by the fundamental theorem on the symbolical 
representation of invariants, a combinant, as far as the coefficients 
a, b, c are concerned, is a rational integral function of determinants 
of the type 



'p hq br 
I Cp Cq Cr 

A like result applies to any number of binary forms. 
Thus for example for two quadratics 

&o^i^ + 2biXiX2 + b2xi, 
a combinant is a function of 

{a^by - aJ}o), (a^b.^ - a^b^), {aoh - a^h), 
as far as the coefficients are concerned. 
But the Jacobian is 

(ao^i — rti^o) ^1 + («0&2 — ^2^0) Xl ^2 + («1^2 — CbJ^l) ^2^ 

and hence a combinant is a rational integral function of the 
coefficients of the Jacobian and the variables. Hence any 
combinant is an invariant form of the Jacobian, and therefore the 
complete system of combinants in this case consists of the Jacobian 
and its discriminant — the latter is equivalent to the resultant of 
the two original forms. 

It is easy to form any number of combinants of two binary 
forms, for 

(i) An invariant or covariant of a combinant is itself a 
combinant, since it is manifestly an invariant form £lnd further 
involves the coefficients of the original forms in the manner 
peculiar to combinants. 

(ii) Let fi and/*2 be two binary forms, /„ an invariant form of 
/i and the corresponding form for Xj/j + X0/2. 

Xi"»/o + ^r-'^Ji + . . . + Xa'^/m , 

then an invariant of this expression considered as a binary form in 
(Xi, X2) is a combinant ofy*! and /g. 



316 THE ALGEBRA OF INVARIANTS [CH, XIV 

For such an invariant is unaltered when we effect a linear 
substitution on the x's because each / is an invariant form ; and it 
is unaltered when we effect a linear substitution on the coefficients 
because it is an invariant of the form in X. written above. 

251. Combinants naturally occur in the discussion of rational 
curves as we shall now shew. 

Suppose such a curve is parametrically represented by 

^, = K- =/, 

^^ = c,^'=fs, (cf§196) 

then the curve is unaltered by a linear substitution effected on 
the x's since its equation is found by eliminating the x's. 

Now if a set of points on the curve be defined by some 
projective property the equation giving their parameters is 
derived from /i, /g, fa in a definite way, hence if by means of a 
linear substitution/i./a^/s become// .//./a' the transformed equation 
for the parameters is derived from//,/',/' in the same way as its 
original form was derived from/, /a,/. 

Thus if the equation be C = it follows that C is a covariant 
of/,/, /s. 

Next, keeping the parameters fixed, to change the triangle of 
reference we replace ^i, ^2, fs hy linear functions of themselves, so 
that/,/,/ are replaced by linear combinations of the form 

hfi + 4/ + hA- 

Now the equation giving the parameters of the set of points 
must be independent of the triangle of reference, for such points 
depend on the curve itself, and the parameters of every point of 
the curve are unchanged when we alter the triangle of reference ; 
hence G is not only a covariant but a combinant of the forms 
/, /, /, and the rational curve is the natural geometrical 
representation of the system of combinants. 

The curve can be equally well defined by the system of 
forms apolar to /, /, /, because these determine /, /, /, 
and the projective properties of the curve are also given by the 
combinants of the apolar system of forms. 



250-252] APOLARITY 317 

We are therefore led to the theorem that the combiuants of 
two apolar systems of forms are identical, and in fact a rigorous 
algebraic proof of its truth will be found in Meyer's Apolaritdt, 

§ 11. 

As an example the reader may verify that in the quartic curve 

gi =' Clx > 6 2 ^^ '^x > S3 ^^ ^a; > 

the points of inflexion are given by 

(he) (ca) (ah) a^hiCx = 0, 
and that, if d^;* and e^* be two forms belonging to the apolar system, 
they are also given by 

{de)dx^ex' = 0. 

The first equation follows from the ordinary methods of the 
differential calculus — the second from the fact that the conditions 
of collinearity of four points are easily expressed by means of the 
apolar system ; if A, be a point of inflexion, and fj, the point in 
which the inflexional tangent meets the curve again, we have 
d^^d|J, = 0, e^^e^ = so that /jl may be eliminated. 

The full discussion of the theory of combinants would lead us 
too far from the methods of the present treatise, and accordingly 
Ave shall content ourselves with the explanation of a "translation- 
principle" connecting the combinants of binary forms with the 
covariants of ternary forms. 

252. It will be convenient to change the notation and to 
suppose a rational curve given by 

^j = ttjiCji + nb^.%\''-^cc,^ + + hx^''] 

^2 = ^2^1" + nh-iCc^''-^ 0:2+ + k^x^^'i- 

^3 = as^i"" + tAx^^'-^x^ + + Ic^x^""] 

Consider the problem of finding the locus of the point of 
intersection of two straight lines which meet the curve in two sets 
of yi points given by binary forms for which a certain combinant 
is zero. 

Let the two lines be 

Wf = 0, v^= 0, 

and denote by ^ their point of intersection. 

The two binary forms are 

aw^i" + nbuXi''--^x.^ + . . . + A;„a;/' 

and a^x^"^ + nh^x-^"^-^ x^-^- ... + ky x^ , 



318 THE ALGEBRA OF INVARIANTS [CH. XIV 

and any combinant is a function of determinants of the type 

Clu Om 

Now UuK -a^bu = (ab^), 

hence the equation of the locus is found by changing (ab) in the 
expression of the vanishing combinant into (a6f ). 

For example if two lines meet the cubic curve 

1^1 = aiX^^ + 2)b^x^X2 + ^c^x-^x^ + d^x^ 
etc. 

in two apolar sets of points we have 

{audy, — a^dy) — 3 {buCr, — b^Cu) = 

or. {ad^)-S(bc^) = 0, 

and hence the locus of their common point ^ is a straight line. 

It is evident that if J. be a point of inflexion then the tangent 
at A and any line through A satisfy the conditions of the problem, 
so that all the points of inflexion of the curve lie on this straight 
line. 

As a second example let us find the equation of the cubic. 
Here the two straight lines meet on the curve and the vanishing 
combinant is the \ eliminant of the two binary forms. 

Following Bezout's method, the eliminant of 

pxi^ + qx^x^ + rx-^x^ + sx^ 

and "p'x^ + q'x^x^ + r'x-^x^ + s'xi 



IS 



= 0, 



W) (K) (p«') 

(_pr') (ps') + {qr') (qs') 

(ps) {qs') (rs') 

and hence making p = au, q = 36m etc. the equation required is 

3(a6|) 3(ac^) (ad^) =0. 

3 (ac^) (ad^) + 9 (bc^) 3 (bd^) 

(ad^) S{bd^) S(cd^) 

It is clear that a similar method applies to the curve of the 
nth degree. 



CHAPTER XV. 

TYPES. 
253. It was proved in § 35 that the effect of operating with 

on a covariant <I> of a simultaneous system of binary forms, which 
includes ag." and bx^ where 

a^"" = (Ao, ^1, ... AnJ_oc^, os^T 

bx'' = (Bo,B„ ... Bn^x„x.^», 

is itself a covariant of the system. 

All covariants thus obtained from <t> are said to be of the same 
type as <I>. In other words two covariants are said to be of the 
same type if one of them is obtainable from the other by means 
of operators of this kind. For example the invariants 

of two quadratics /, cf) are all of the same type. 

It should be noticed that this connection between two co- 
variants is not necessarily reciprocal; two covariants <I>i, <I>2> 
where <E>2 is obtainable from <I>i by operators of the required 
kind, are of the same type, even if ^j is not so obtainable from 
^2- Thus if F(a, a', ...) is a simultaneous covariant of a system 
of quantics which includes /=««'* = aVS (f> = ba;^, and if F is of 
the second degree in the coefficients of / but does not contain 
those of (j>, the covariant 

F(a, b, ...) + F{b, a, ...) = (<|,|.) F {a, a, ...) 



320 THE ALGEBRA OF INVARIANTS [CH. XV 

is of the same type as F{a, a', ,..) ; but here 

F{a,a,...)={^^F{a,h,...) 

and we see that F{a, a', ...) and F{a, b, ...) are of the same type. 

It will be seen in this way that two covariants ^i, <E>2 may 
each be of the same type as a third covariant <I>, although neither 
<J>i nor <l>2 is obtainable from the other by an operator of the kind 
considered. In view of this the further statement is necessary 
that covariants which are each of the same type as a third 
covariant are (by definition) of the same type as each other. 

254. Every covariant of degree m, of one or more quantics, is 
of the same type as a covariant which is linear in the coefficients 
of each of m quantics — the number of quantics in the system 
being, of course, increased if necessary. Any such representative 
covariant is, for convenience, called a type; a type is then a 
covariant which is linear in the coefficients of each of the 
quantics concerned, it being understood that these are not special 
quantics of the system and that the word type is used in a purely 
formal sense. 

Thus for three quadratics (§ 139 a) 

{ab) (be) (ca) 

is an irreducible type, and furnishes only one invariant of the system ; (ab)^ 
is also an irreducible type and furnishes six invariants. 

It should be noticed that if fi,f2,fz are the quadratics, and 

the invariant i'^i.iifzf' 

is not of the same type as (a&)^, because J^^^ ^^ "^^ ^"^ '^^ ^he fundamental 
quantics of the system. 

Consider the covariants of a simultaneous system of binary 
forms of the same order. When the number of binary forms is 
indefinitely increased, the number of irreducible covariants will 
also be increased without limit ; in fact the number of irreducible 
covariants belonging to any one type will be indefinitely increased. 
The question arises — does the number of irreducible types increase 
indefinitely too ? This question has been answered in the negative 



253-256] TYPES 321 

by Peano*. Peano's theorem is the following: Every type of a 
system of binary n-ics which does not furnish irreducible covariants 
for a system of n n-ics is reducible, with the single possible ex- 
ception of the invariant type 

A^ Ai ... An 

Bo -Bi ... Bn 



Ko Ki ... K,, 

where {A^, A^, ... A n\x^ , x^Y 

(Bo,B^, ...Bn\x„x,y 



{Ko,K^, ...Kn\x„x^f 

are ?i + 1 n-ics. But if this invariant is reducible, all types are 
reducible which do not furnish irreducible covariants for n — \ 
n-ics. 

A proof of this theorem is given in the next chapter. 

255. As was pointed out in § 21 there are two principles 
by means of which the reduction of a covariant has to be 
attempted, viz. : 

(i) The fundamental identities 

(6c) (ad) + (ca) (bd) + (ab) (cd) = 

(be) a^ + (ca) b^ + (ab) c^ = 0. 

(ii) The fact that the interchange of two symbols which 
refer to the same quantic does not alter the actual value of a 
symbolical product. 

To effect the reduction of a type the first of these two 
principles must alone be employed. 

256. The quadratic types. The quadratics will be denoted, 
as usual, by 

^x'j ^x'} ^x'y • " • 

* Atti di Torino, t. xvii. p. 580 (1881). See also Jordan (Liouville, 1876, 
2 S6r. III.), who proved that the number of irreducible types belonging to any 
simultaneous system of forms, the order of each of which is less than some fixed 
number n, is finite. 

G. & Y. 21 



322 THE ALGEBRA OF INVARIANTS [CH. XV 

For invariants the only symbolical products to be considered 
are 

{ahf 

^ {ah) (6c) {ca) 

{ah) {he) {cd) {da) 



The first two of these are irreducible, for the fundamental 
identities give us no relations by which we may reduce them. 

The other invariant types are all reducible. For 

2 {ah) {he) {cd) {da) = {ahf {cdf + {hcf {daf - {acf {bdf ; 

operate on this identity with 

A A 

then 

{ah) (he) {cd) {de) + {eh) {he) {cd) {da) 

= - {cdf {ah) (he) - {hcf {ad) {de) + {hdf {ae) {ce). 
But 

{ah) (be) {cd) {de) - {eh) {he) {cd) {da) = (be) {cd) {dh) {ae). 

Hence 

2{ab){hc){ed){de) 

= {he) {cd) {dh) {ae) - {cdf {ah) {he) - {hcf {ad) {de) + {hdf {ae) {ce). 

By means of these two identities all the invariant types of 
degi'ee greater than three are at once reduced. Now any 
covariant of a system of quantics of even order must be itself 
of even order (§ 20) ; hence any covariant type of the quadratic 
may always be obtained by replacing one or more letters in 
the symbolical expression for some invariant type by the variable. 
For example, from {ah)- we obtain ax' on replacing hi by — iCg and 
bs by a^i. Hence the irreducible covariant types are 

a/, {ab)axhx. 

The quadratic has then only four irreducible types (compare 
§139 A), 

{ahy, {ah){be){ca), 

ai, {ab)axbx. 



256-257] TYPES 323 

The second is in fact the determinant type referred to above, 
for as has already been pointed out, 

(ab) (be) (ca) = — ' ai^ aittg (^2^ 
b,^ bA V 

257. The cubic t3rpes. It is possible to obtain the complete 
system of types for binary forms of a given order by a method 
almost identical with that of Chapter vi. for covariants of 
a single binary form. The reductions in this method are 
generally very difficult to obtain. The cubic types, however, 
can be thus obtained simply. It is thought unnecessary to go 
through the general argument, the alterations in Chapter VI. 
to meet the case of types being mainly verbal. It should be 
noticed that the finiteness of the complete irreducible system of 
types could thus be demonstrated. 

Let a^^, bx\ Cx\ ... be the cubics. The symbol F will be used 
to denote any one of them indifferently. The types of .degree two 
are 

{ab)ax^b^ = J, {ab)- a^bx = H, (aby. 

Consider first the types of grade unity. These all contain 
a factor (ab), and hence are terms of transvectants of J with 
types of grade not greater than unity. 

In fact any such type is a term of a transvectant of the form 



{J,J,,..Jr, F,F,...F,)\ 
where the bar over the left-hand member indicates any type 
obtained by convolution from the product there written down. 
It follows at once that every type of unit grade can be expressed 
as a sum of numerical multiples of such transvectants. Now by 
§§ 74, 75 any type obtained by convolution from Ji, J^, ... Jr 
is of grade two at least. Hence the only irreducible types of unit 
grade are expressible as transvectants of the form 

(JJ,...Jr, F,F,...F,)'. 

If \ = 1 this is clearly reducible — for J" is a Jacobian. 

If \ > 1, this contains a terra of grade two. 

Therefore the only irreducible type of unit grade is J. 

21 2 



324 THE ALGEBRA OF INVARIANTS [CH. XV 

Now the irreducible types of the quadratic H are 

Hence the tj^es of grade two are expressible as transvectants 
of the form 

{H,H,... H,K,K,... Kp, F,F,...FyJ,J,... J.f 

or of these multiplied by invariant types. 

In the first place we notice that K and J are Jacobians and 
hence we may suppose that neither yS nor B exceeds unity. 

We have the following types to consider : 

{H,F), {H,F)\ (H,H,,Fy, {H,HA,F,F,y 

{H,J)\ {H,H„Jy, {HJI,,jy 

(K,Fy, (HK,Fy, (H,H,K,F,F,y, (K,jy, {HK,Jy, {HK,jy. 

Of these {H^H^H^, F^F^y contains the term 
(«! hy {aAy (ashy (aicO (61 Ci) (a^c^) (hc^) (asd) (63C2) 
= (aA) (61 Ci) (CiOi) . (a.262) (&2C2) (CjOa) . {a^bsy (aA) (a^h) (OsCi) (63 C2) 

=i (cha,y (aAy ((hc,y ; 

Aa,y (hAy {hxc,y \(aAy((hbi)(,aA)(chC,){bsC,),...(^71). 

{c,a,y {cAy {c^c,y 

This is a sum of terms obtained by convolution from products of 
four types H, and hence is reducible. In exactly the same way 
the type {H^H^KyF^F.^y may be reduced. 

The type (H, Jy contains a term 

iaby (be) (ad) (cd) c^^d^ 

= - i {oh) {(aby (cdy + (bey (ady - (acy (bdy] c^d^, 

which may be expressed in terms of the type (H^H^) and reducible 
forms. 

The type (K, jy contains the term 
(H,H,)(H,c)(H,d)(cd)C:,d^ 

= - i {(H,H,y (cdy + (H,cy (H,dy - (H,dy (H,cy\ c^d^, 

and hence is reducible. 

The type (HjH.^, Jy contains the term 

(Hu (H^, Jy) = 2(H„ (H„ H,)) + reducible terms, 

and hence is reducible. In the same way the types (HiH,^,jy, 
(HK, jy, (HK, jy may be reduced. 



257] TYPES 325 

The type 

{K, FY = (H,H,) (H,F) {H,F) F, 

= {H,Fy (H„F) H,, - {H.Ff {H,F) H,, ; 
and 2 (H,H„ Ff = {H,Ff {H,F) F^ + {H.jy (H,F) F^. 

Hence both {K, Ff and {H^H.^, F)- can be expressed in terms 

of the type 

{H,F)"'{H,F)F, = L^. 

Lastly, (HK, Ff contains the terra 

(aih,)- {a^Ky (asbsf {h^a^) {(hc) {a^c) {h^c) b^x 

= ((«! hy (O'shY (bi a^) (a, c) (agc) (bsc) a^i, b^i^. 

Now the left-hand member of this transvectant is a type of 
degree six and order two ; looking back at the possible types of 
this order we see that it must be of the form 

The second and third of these terms contain invariant factors, 
and can therefore only lead to reducible terms in the above trans- 
vectant. Also 

2L,L, = 2 (aby (ac) (be) c, . {dey (df) (ef)/^ 

= (ab)ide)Cxfx (ady {aey {a/y 

(bdy (bey (b/y 
(cdy (cey (cjy 

= S (H,H.;)' H, + ^ (H,H,) {HA) H,,H,, 

= S (HiH^y H3. 

The type (HK, Fy is thus reducible. 

The complete system of irreducible cubic types is then 

(aby, (aby (be) (cdy (da), (aby (be) (cdy (de) (e/y (fa), 

(aby (ac) (be) c^, (aby (edy (ae) (be) (ee) 4, 

(aby aj)^, (aby (be) (edy a^d^, 

ax, (aby (be) a^ex, 

(ab) a^bxy 
there being ten types in all. 

The system of irreducible concomitants for two cubics may be 
obtained from the system of types or else directly, they will be 



326 THE ALGEBRA OF INVARIANTS [CH. XV 

found in the works of Clebsch and Gordan, The syzygies between 
them have been obtained by von Gall {Math. Ann. Bd. xxxi.). 

258. Perpetuants. The irreducible seminvariants (§ 32) of the 
binary form of infinite order are called perpetuants. The complete 
system of perpetuants for one binary form of infinite order has 
been obtained by Macraahon* and Strohf. The system is, of 
course, infinite in extent, but the individual members of it have 
all been identified. 

The complete system of perpetuants for any simultaneous 
system of binary quantics of infinite order was obtained by 
Macmahon;}:; and in particular the perpetuant types may be at 
once obtained from this paper. 

The method by which these results were obtained, does not 
fall within the scope of this book. The results have been 
obtained more recently by means of the symbolical notation 
which has been here developed; and this investigation § we 
shall follow. 

259. A covariant is completely defined when the determinant 
factors in its symbolical expression are known ; it will be 
convenient to use this part of the symbolical expression only. 
In dealing with forms of infinite order, it must be remembered 
that the complete expression for a covariant contains each of the 
factors a^, h^, ... raised to an indefinitely high power. 

The identity 

(6c) ax + (ca) h^ + {ah) Ca; = 

may now be written 

{he) + {ca) + {ah) = 0. 

By means of this identity any factor (6c), in a covariant, which 
does not contain a may be replaced by 

{ac) — {ab), 
i.e. by factors which do contain a. Thus all covariant types may 
be expressed in terms of those which are of the form 

{ahf{acY{ady... 
where a is any one of the letters chosen at will. 

* Proc. Lond. Math. Soe,, vol. xxvi. See also Avi. Journal, vols. vii. viii. 

t Math. Ann., Bd. xxxvi. 

t Camh. Phil. Soc. 'Trans., vol. xix. pp. 234 — 248. 

§ Grace, Proc. Lond. Math. Soc, vol. xxxv. 



257-261] TYPES ^^^ 327 

The types of this form are all linearly independent, for no 
linear algebraical identity can connect their symbolical expres- 
sions. 

Hence if all reducible types were expressed in terms of types 
of this form, we should be able to write down the perpetuant 
types. 

It must be remembered that a is a perfectly definite quantic 
of the system. Further that the remaining quantics concerned in 
any particular covariant will be considered in a particular order 
determined beforehand. 

260. Consider the types of degree three ; if w be the weight, 
we know that 

{hcY = {(«c) - {ah)]"" 

= {acf" - w {ah) (acy-' + + (- 1)^ («&)«'. 

Hence the covariant (ab) (ac)'^~^ is expressible in terms of reducible 
covariants and of covariants in which the index of (ab) is greater 
than unity. Hence all perpetuant types of degree three are 
expressible in terms of the types 

(ab)^{acy, \<t:2, fi^l. 

It should be noticed that of the three quantics concerned any one 
may be chosen to correspond to a, b or c respectively. 

Further, the only reducible covariants of degree three and 
weight w are represented by 

(6c)% (aby, (ac)"", 

and hence the seminvariants (aby {acY (X. -"^ 2, /t -^ 1) are both 
independent and irreducible. 

261. Types of degree four may all be expressed in terms of 
the independent forms 

{abY {acY {ady. 

If \ or /Lt be less than 2, then as in the previous paragraph the 
index of (ab) or (oc), as the case may be, can be increased at the 
expense of the index of (ad). 



328 THE ALGEBRA OF INVARIANTS [CH. XV 

Thus, since the reducible covariant 
{aby {cdy-^ = {ahf [{ad) - (ac)}«'-^ 



+ (^ 9 ) («&)' («c)^ (ad)''-^-' -..., 



the covariant {ahy (ac) {ady''~^~^ can be expressed in terms of 
reducible forms and of covariants 

(abf (acY {ady-''-^ (/^ > !)• 

When both \ and //, are greater than unity, say \ + /jb = M, we 
may express, by means of Stroh's series § 64, the products 

(aby (acY 

in terms of the following three sets : 

(i) {ab)^,(ab)^-'(ac),...{aby{ac)^-*; 

(ii) {ac)^,(ac)^-'(ab); 

(iii) (bc)^, (bc)^-' (ab). 

The products contained in (ii) and (iii) need not be considered, 
for the corresponding covariants can be expressed linearly in terms 
of reducible covariants and of covariants in which the number 
of factors involving a, b, c only is greater than \ + fi. These 
latter forms can be dealt with in the same way. 

Thus we see that ultimately we can express all the covariants 
of degree four in terms of reducible covariants and of such as 
have the factor (ab)*. Further we have seen that we may suppose 
the coefficient of (ac) to be greater than unity: hence all covariants 
of degree four can be expressed in terms of reducible covariants 
and of covariants of the form 

(aby {acY (ad)" 

where X-«j^4, fi-^2, v-^1, and the arrangement of the letters 
a, 6, c, d has been fixed beforehand. 

262. The theorem can now be proved in general by induction. 
We shall assume that all covariant types, of a system of binary 
forms of infinite order, which are of degree n + 1 or less, can be 



261-262] TYPES 329 

expressed linearly in terms of reducible covariants and of covariants 
of the form 

(aoi)^' {aa^Y^ . . . (aa„)*'' , 

where \<^'^\ \2H:2""', ... Xn<i:l, 

and the arrangement of the letters a, a^, cto, ... a„ is fixed. 

For degree ?? + 2 we have only to consider the covariants of 
the form 

(aoj)^! (aa-.y^ . . . (aa,i+i/''+' = (aa^^y^ {aa^^-^ R. 

Now {aa^^R is of the same symbolical form as a covariant 
of degree w + 1 of the system ; hence, using the result for that 
degree, we may, if Xa < 2'*~^ express it in terms of covariants of 
the same form but for which the index of {aa^) is not less than 
2"~\ and of reducible covariants. In the same way,- if Xj < 2'*~\ 
the index of {aa^ in the product (aoi)^' R can be increased. 

Let Xi + X, = J/ <{: 2 . 2"-S N = 2"-^ ; 

then, as before, by means of Stroh's series all products 
(aai)^ {aa.^ can be expressed in terms of the M-\-l following 
products : 

(i) {aa,yi, {aa,yi-^ {aa,), . . . {a(hf^{aa^) ^^-^, 

(ii) {aa,y^, (aa,yr-^ (aa,), . . . (aa.y^-^-'-' (aa,y-\ 

(iii) (a, aoy^, {a,a,y^- 1 (aa,), ...{a, a,y^-^+'- (aa^)^-\ 

The products contained in (ii) and (iii) need not be considered, 
for the corresponding covariants have factors of the [form {aa^R 
where p < 2"~^ ; hence these covariants can be expressed in terms 
of reducible forms and of products which contain a greater 
number of factors involving a, a^, a., only. 

The products contained in (i) all contain the factor {aa-^^. 
Hence all covariants of degree n + 2 are expressible in terms of 
reducible forms and of covariants which have the symbolical 
factor (aaiY^. But these can, by an application of the assumed 
result for degree n + 1, be expressed in terms of reducible forms 
and of the covariants 

(aa^Y^iaa^y^ ... (aan+i)^"+\ 
where Xi^2", \<i;2»-\ ... X^+iH:!. 



880 THE ALGEBRA OF INVARIANTS [CH. XV 

The theorem is then true for degree n + 2 if it is true for 
degree n + 1; it has been proved for degrees three and four and is 
therefore true for all degrees. 

263. It should be noticed that it has not been proved that the 
covariants i^etained are irreducible. It is practically certain that 
this is so, but no rigorous proof has yet been given. The number 
of covariants retained which are of degree n + 1 and weight iv 
may be found as follows. If 

i.e. if 'iv<2^^ — l, all the covariants are reducible. If w*i;2'^ — l, 
the covariants retained are of the form 

(aaiY" (aao)-"" . . . (acin) . R, 
where R is any product 

(aai)^' (aa,/^ . . . (acinY" , 
and Xi + X2 + ...+X„ = w-2" + l. 

Hence the number required is the coefficient of ii;'*'-2"+i in the 
expansion of (1— ic)~" — this being the number of homogeneous 
products of dimensions w — 2" + 1 of n letters. This is equal to 
the coefficient of a;*'' in the expansion of 

This generating function for perpetuant types is the same as 
that obtained by Macmahon's methods. 

264. The results thus obtained for perpetuant types are of 
great use in obtaining either the types or the ordinary covariants of 
a binary form of finite order. All that was required in the course 
of the argument was that the weight of the covariant under 
consideration should not exceed the order of the quantic — or 
quantics. Thus any covariant of weight w and of degree h of the 
binary n-ic which is such that 

2«-i -\>wi^n, 
is reducible. 

In § 114 the system of forms A^ for a single binary form of 
order ^12 was discussed. The above considerations of weight, 



262-265] TYPES 331 

alone shew that the following forms which were there retained 
are reducible 

{aby {hcf (cdf (de), (ahf (bcf {cdf {de\ {ahf (be) (cdy (deY ; 

the argument applies to the last of these three covariants only if 
the order of the binary quantic «a;" is greater than 12, it will be 
seen later that this is indeed reducible for the 12-ic. 

The remaining forms of the system would not be reducible as 
perpetuants and hence we cannot hope to reduce them for forms 
of finite order ; the two forms 

(aby (bey (cdy, (aby (be) (cdy, 

however, are congruent mod. (aby, as will be presently proved. 

265. The theorem for covariants of forms of finite order 
corresponding to that which has been proved for perpetuant types 
is the following*. 

All covariants which are of the first degree in the coefficients of 
each of the quantics 

can be expressed linearly in terms of 

(i) covariants ofthefoi-m 

(flja.)^! (a.as)^- ... (as-ias)'^^, 

where \i ^ 2«--, \, ^ 2«-s, ... \s^l, 

and the arrangement of the letters a^, a.^, ... as is fixed ; 

(ii) covariants which have a symbolical factor 

(aj,akf(akai)'"''-^; 

(iii) products of covariants of lower total degree. 

The proof of this theorem follows that for perpetuants very 
closely. We first assume that it is true Avhen the total degree of 
the covariant considered is not greater than S — 1, and prove it 
when this total degree is S. 

Now the covariants to be considered can be expressed in terms 
of transvectants 

(a,;^,G,_,Y (I), 

• Young, Proc. Lond. Math. Soc. 1903. 



332 THE ALGEBRA OF INVARIANTS [CH. XV 

where Cj-i is a covariant of the first degree in the coefficients of 
each of the quantics a^ **', a, ^^, ... as "« . 

On the assumption made (7s_i can be expressed linearly in 
terms of covariants of the second class, of covariants of the form 

and of products of covariants of lower total degree. 

If Cs-i is of the second class the transvectant (I), and each of 
its terms, must be of the second class. 

If Cj_i is a product of two covariants P, Q, then the total 
degree of P being less than S we may express it in terms of 
covariants of the form 

{ahY^{hcY^...{fgyi, 

and of covariants of the second and third classes. 

If /Lt + /ii be greater than the order of the form a the trans- 
vectant contains a term of the second class ; if it be not greater 
than this order the transvectant contains a reducible term. 

If P is of the second class the transvectant itself belongs to 
the second class ; and if P is of the third class, we may take one 
of its factors and proceed as before. 

If C5_i belongs to the first class, we may take 
C5_i = {a^(hY^ (asaiY' ■ • • (^s-i ^sY^ , 
then when Xs + M -^ ?^2, the transvectant contains a term belonging 
to the second class ; but when Xg + /"■ < n^ it contains the term 

{a^a^Y (aatts)^" {azCuY' ••• (a«-i«6)^«, 
and hence covariants of this form alone need be considered. 

266. Let us now, for the sake of shortness, write 

Then we shall proceed to shew that, if /c < 2*"=', the trans- 
vectant 

{{a^a^Y ((^2a3Y, a^'^'-^^q/Yy^x 

can be linearly expressed in terms of covariants of the second and 



1 



265-267] TYPES 333 

third classes, and of covariants which contain a greater number of 
factors involving aj, a^, a-i only. 

The two sets of covariants (aiCt-iY (a^a-iY, and {(a^a-iY, '^s/O''* 
where /* + /c has a fixed value and k takes all possible values less 
than 2^""-', are equivalent. 

Hence if (aia^Y = ax"'^"-""", 

the above transvectants may be linearly expressed in terms of the 
following, 

Any one of these transvectants is a covariant of unit degree in 
the coefficients of each of the 8—1 quantics 

it can therefore, by hypothesis, be expressed in terms of covariants 
of the form 

(aos)'^' {a^aiY' ■•• {a^-iCi^Y^-i, 

where fi, ^ 2«-^ /., ^ 2«-^ . . . fi,_, ^ 1, 

and of covariants belonging to the second and third classes. 
Thus the number of factors involving aj, a^, a^ only, can be 
increased when k < 2^~^ 

It should be noticed that covariants of the second class here 
include those which contain the factor 

(a„a)"'(aa;k)"'+"-^-2'^-''. 

It is easy to see that such a covariant belongs to the second 
class in the enunciation ; for, we may suppose 

v^ iii-'r n,, — 2/i — J/, ni ^ n.2, 

and therefore v'^iii — fx, 

the covariant considered then contains the factor 

267. The covariant 

(a^a^Y («2«3)^' {c-iaiY" ••• {as-iasY^-i 
is a term of the transvectant 

({a^a.Y (a^a^Ys aC^'q/Y'y^x, 
and hence differs from the whole transvectant or from anv one of 



334 THE ALGEBRA OF INVARIANTS [CH. XV 

its terms by covariants in which the number of factors involving 
Oi, a2, fts only is greater than \^ + fi. 

By § 266, we see that we may suppose that neither X, nor fi is 
less than 2*~^ ; and hence that Xo + /* -"t 2^-\ The covariant 

can be linearly expressed in terms of the covariants 
(i) (a:aO^^+^ ia,a,Y^+>^-Ha,a,), ... (a,a,f-'{a,a,y^^'^-^'-\ 
(ii) (a,a3y^+^ (a,a,y^+'^-' (a,a,), ... (a,asY^+>^-^'-'+^ {a,a,f-'-\ 
(iii) (a,a3)^^+^ ia,a,Y^+'^--'{a,a,), ... (a,a,)^+>^-'^'-'+^asay-'-\ 

Transvectants of a covariant from one of the last two rows with 
aj^*~^*qy'' can be expressed in terms of covariants which contain 
a greater number of factors involving aj, ag, as only. Hence we 
may ultimately express all covariants in question linearly in terms 
of covariants having a factor (aitto)^', where Xj -^ 2*"^, and of 
covariants belonging to the second and third classes. Proceeding, 
as in § 266, with the covariants which have a factor (wia^)^' where 
Xi -^ 2*~-, we see that all covariants may be expressed linearly in 
terms of covariants of the form 

(Oiaa)^' (ttaaa)^ ... (as-ias^s, 

where X^ ^ 2^-\ \, ^ 2^-^ ... X5 ^ 1, 

and of covariants of the second and third classes. 

Thus the theorem is true when the total degree of the 
covariant is 8, provided that it is true when this total degree is 
less than B. 

268. It remains to shew that this theorem is true when the 
total degree is three. 

The covariants to be considered are 

(aittj)*' (Oaaj)^ (asOi)'*'. 

Unless Xj + X2 + X3 is less than each of the numbers Wj, n.^, n^, 
this covariant belongs to the second class. For let 

Xi + X2 + X3 ^ ?l2, 

then by means of the identity 

(astta) = - (ctiaa) - (a2«3) 



267-270] TYPES 335 

the above covariaut can be expressed in terms of the forms 

and belongs to the second class. 

If \i + X2 + ^3 is less than each of n^, lu, 7I3 the argument used 
for perpetuants may be repeated here word for word. The theorem 
is then true for total degree three, and therefore for any total 
degree. 

269. If all the quantics are of the same order n we obtain a 
theorem concerning covariant types of a simultaneous system of 
binary n-ics. 

In this case the covariants of the second class contain a factor 
of the form (aby {bcy-'^, and hence a factor of the form 

(aby (bcy-i' (cay, 

where '^ "^ 1 ' /3 "i 2?i — 3/a, 

(see § 68). 

Hence all covariant types of a system of binary n-ics can be 
expressed linearly in terms of 

(i) Covariants of the form 

(ttj a,)^! {aM^y^ . . . (ag_i asys , 

where \ ^ 2^--, \, <j: 2^-^ ... X^ ^ 1, 

and the order of the letters is fixed beforehand, 

(ii) Covariants tuhich have a factor ofthefonn 

(aby (bey -^ (cay, 

where '^^ ^> p-^2n — S\. 

(iii) Products of covariants of lower total degree. 

270. The theorem just proved expresses all irreducible co- 

variants, of grade ^ ^ , in terms of a certain number of forms, 

which, there is good reason to believe, are irreducible when ?i is 
infinite. If this be so, these forms are certainly irreducible for 
finite values of n. 



336 THE ALGEBRA OF INVARIANTS [CH. XV 

However, it does not follow that we cannot express them in 
terms of covariants of higher grade or else of covariants of the 
second class. 

In this connection we shall prove the following : all covariant 
types of the binary n-ic can be expressed linearly in terms of 
covariants of the form 

where \ ^ 2^2, 7^ > Xj, X3 > Xj, ... X^^^ > ^s-i, 

and of covariants belonging to the second and third classes. 

Using the previous theorem we see that covariants of the form 
G = (aitta)^' (aattay^ • • • (cis-iasYs-i 
alone need be considered. 

Let /*! :^ 2fi2, then (7 is a term of 

((flhOa)^" (a^a^y^ {a^a,Y^ ... (aa_xafi)^a-ia,^«-«*a5^» -«*-«»... )%=<,, 

and differs from any other term by covariants which involve a 
greater number of factors containing aj, a^, as only. 

Then by Stroh's theorem, we may express (oia^Y' {a^f^aY^ i^ 
terms of covariants of the form 

{abY' {bc)\ 
where \i-^2\2, \i + X2 = /*i + At2j 

and a, b, c are the letters a^, a^, a-j in some order. 

Let (abY' = a^^-^s 

we have then to consider covariants of the form 
(aas)*^ (oaa^Y' • • • {as-iasY^-i. 
If X2 < /Its we may consider the transvectant 
{{oia;Y'{(ha,Y'> (a,aeY' .•• (as-.asys-i a./'-'^ni, "-'^^''^^Y^y-x- 

Then (aaa)^ (asaiY' can be expressed linearly in terms of 
(agai)^"*''*', and of members of the sets 

{aasY {asa,)^+''*-', {m,Y {aiasY'+ '''-'', 
where k > — ~-^ . 



270-271] TYPES 337 

We may proceed in exactly the same way at every step, and so 
prove the theorem. 

It will be noticed here that the order of the letters in the 
covariants 

is not fixed. 

271. The Maximum Order of a covariant. Let us con- 
sider the covariant types of a system of quantics of which none of 
the orders exceeds n. By § 265, these types can all be expressed 
in terms of covariants of three kinds. Consider the covariants of 
the second kind. These contain a factor of the form 

We may suppose that X -^ n^ — X; the order of the covariant 

is then /ii + ^i, — 2X, :^ Wi < n. 

Now let us introduce a new symbol, for each covariant whose 
order does not exceed n. Covariants of the second kind are thus 
at once reduced in degree. Covariants thus reduced may them- 
selves be expressed in terms of covariants of the three different 
kinds. The covariants of the second kind may again be reduced, 
and so on. Hence finally we have expressed the system of 
covariants in terms of covariants of the form 

(aitts)^' (a^ttsY^ ... (as-ias^s-i, 
where \,^ 2^--, \^ '2^-\ ... Xs-i^l, 

— tti "', ttj "s ...as "« being either members of the original system 
of quantics or covariants of that system whose order does not 
exceed n — ; and of products of covariants of lower degree. 
The covariant of maximum order must then be of the form 

where X, < 2«-^ X, < 2«-^ ... Xs-i < 1, 

and ni, n^, ... ns are all equal to or less than n. 

The order of this for a given value of 8 is a maximum when 

Xl = 2«-^ \.3 = 2«-^ ... Xa_i=l, 

ni = ?i„ = . . . = Hs = n. 

G. & Y. 22 



338 THE ALGEBRA OF INVARIANTS [CH. XV 

In this case the order is 

nS - 2 (1 + 2 + ... + 2«-2) = 7iB -2^ + 2. 
The maximum order is then the gi-eatest of the numbers 

n, 2n-2, Sn-6, ... w8 - 2* + 2, .... 

It is easy to see that if n = 2^ + ni, where ?ii<2^ then this 
maximum order is 

(\ + l)(2^ + w0-2^+i + 2 = (\-l)2^ + ?i,(\ + l) + 2. 

Comparison with perpetuants shews at once, that if the results 
for these are absolutely accurate, then the maximum order just 
obtained is always reached — even for a single quantic of order n 
except for the case n = S. 

Ex. (i). The covariant (ab)^ (be) (cd)* {def of the twelvic referred to in 
§ 264 is of weight 13 and hence must be reducible to co variants of the second 
and third classes ; it is evidently reducible in the usual sense of the woi^d. 

Ex. (ii). Shew that the following covariants of the ten-ic can be expressed 
in terms of reducible covariants and of covariants of higher grade : 

{ahf{hcf{cd), 

{ahf{hcf{cdf, 

{ahf{hcf{cdY{de). 

References to papers by Jordan and Sylvester on the problem of § 271 and 
allied problems regarding weight and degree will be found in Meyer. The 
limits hitherto given are much too high for large values of n. 



CHAPTER XVI. 

GENERAL THEOREMS ON QUANTICS. 

272. In this chapter certain results are obtained by an applica- 
tion of the theory, or rather the notation of the theory, of finite 
substitution groups. So little knowledge of this subject is required, 
that, for the sake of readers unacquainted with it, we shall start 
from the commencement, and prove the few well-known theorems 
required. 

In the first place a function of 7i variables 
is under consideration ; the function 

J \^2, ^1 J ^3, • • • ^n) 

is derived from this by the interchange of the two variables x^ and 
Xo. The operation by which the latter function is obtained from 
the former is called a substitution, it is usually denoted by the 
symbol (xiX.2). Thus we may write 

J (X^, Xi, Xg, ... Xji) ^ {XiX^}/ {^Xi, Xo, X^, ... Xyi). 

A more general example of a substitution is the operation by 
which the arrangement of variables 

Xi , Xo , • . . Xji 

is changed to 3/1' 2/2, •.• 2/n, 

the y's being the variables Xi, x.2,...Xn arranged in some order. 

This substitution is often written 

(Xi Xo . . . fl/jj N ^ 
2/1 2/2. ..2/n/' 

*^"^ U 7"'T) f^^" ^2' ••• ««)=/(2/i' 2/2. ••• 2/n)- 

\</i 2/2 •• • yn/ 

22 2 



340 THE ALGEBRA OF INVARIANTS [CH. XVI 

Substitutions which represent merely the interchange of two 
variables are called transpositions ; thus the substitution {xiX^ 
introduced above is a transposition. 

The product of two substitutions. Let Si, s^ be any two sub- 
stitutions of the letters w^, x^, ... Xn, the meaning here attached to 
the product s-^s^ is that it is an operation which when applied 
to a function of x-y, x^, ... Xn is equivalent to the operation first of 
■S2 on this function and then of Si on the resulting function.- (The 
usual convention is that the substitution on the left is the first to 
operate, but the above is more convenient for our present purpose.) 
Thus 

The effect of s^ is merely to produce a rearrangement of the 
variables, the effect of s^ on the resulting function is to produce a 
fresh rearrangement; thus the product of two substitutions is a 
substitution. 

It will be seen at once that substitutions obey the distributive 
law, for 

Si [SiS^\j \Xi, X^i ... Xn) = Si [SnSsJ \Xi, X^, ... X^)} 
= L51S2J S3 J yXi , Xo, ... X,,i). 

On the other hand substitutions are not in general commutative, 
for example : 

\X1X2) \XiXi) J \Xi , X2, i^s) '^ \^i^a)j \^s> ^2> '^i) 
~y \^3> ^i> ^2)) 

^^y \*'2> "^S) •^]/* 
Any substitution can be represented as a product of trans- 
positions. 

For any rearrangement of the letters Xy, Xr,,...Xn can be 
produced, first by an interchange of Xi and one other letter by 
which Xi takes its new position, next by an interchange of x.^ and 
another letter by which x^ is brought to its new position, and so on. 

It will be found that a substitution can be represented as a 
product of transpositions in a great number of ways, e.g. 

\X1X.2) ^ (^2'^3^ V'^l*^3/ V'^2'''37 5 



272] GENERAL THEOREMS 341 

but the numher of transpositions in a product which represents a 
given substitution is ahuciys even or always odd. For consider the 
function 



A = 



1 
1 



— li \Xf Xg) J 



1 



r, s 



the effect of any transposition operating on A is merely to change 
its sign. Hence 

sA=+A 

according as 5 is a product of an even or odd number of sub- 
stitutions. Substitutions will be called even or odd according as 
the number of transpositions of which they are composed is even 
or odd. 

Consider any rearrangement of the letters x^, x^y.-Xn', let Xr 
be the letter which takes the place of x^ ; Xg that which takes the 
place of Xf] xt that which takes the place of Xg, and so on; we must 
sooner or later arrive at a stage when x^ is the letter which takes 
the place of x^^ the last of the series. The substitution which 
replaces x-i^ by a-v, ^V by Xg, Xg by Xt and so on, and finally x^^ by x^ 
is usually written {x^^XrXgXt . . . Xy) and is called a cycle. It is 
evident from the definition that 

\XiXyXgXi . . , Xn) ^ yXfXgXi , .. X'uXi). 

The rearrangement considered may be produced so far as the 
letters in the cycle are concerned by operating with {x^x^XgXt . . . Xu). 
Let Xa be one of the letters not contained in this cycle, then we 
may suppose that x^ in the new arrangement takes the place of x^ 
and proceed as before. Thus we see that the rearrangement may 
be produced by operating with a number of independent cycles, 
i.e. cycles such that no two contain a common letter. Hence any 
substitution is equal to a pr'oduct of a number of independent cycles. 

That operation which leaves any function operated on un- 
altered is called the identical substitution and is written 1. The 
product 

\x-^x^) . yXiX^) = yXiX^) 

leaves every function unaltered, hence 

yX-^X^)' ^^ 1. 



342 THE ALGEBRA OF INVARIANTS [CH. XVI 

Consider the rearrangement of the letters Wi, x^,... Xn produced 
by any substitution s ; there is a perfectly definite substitution 
which will change this new arrangement back to the old arrange- 
ment. This is called the inverse substitution of s and is written 
s~^. In virtue of the definition of s~^ we see that 

8 Sj \Xi , X^, ... Xfi) =J (Xi , X^, , .. X^)) 

and hence s~^s=l. 

Again it is to be observed that the result of operating on the 
new arrangement of the letters wdth ss~^ is to leave it unaltered, 
hence 

SS-' = 1. 

Consider the powers of any substitution, 

S, S", S^, ... ', 

they are all substitutions, and since the number of different sub- 
stitutions of n letters is finite — in fact nl, the number of possible 
arrangements of those letters — these powers cannot be all different. 
Hence for some values of h, k, 

In virtue of the associative law, we can write 

o2om __ gl+m 

Hence s^"^^ = s*+'. 

Further, whatever substitution <t may be, 

o- . 5^ = o-s* ; 
hence if o- = (s~^)^ we see that 

and hence 1 = ^~^. 

We may suppose that k > h, hence among the positive powers 
of s we must find the identical substitution. Let p be the smallest 
positive index for which s^ = l, then^ is called the order of the 
substitution. 

The substitution a-sa~^ 

is said to be conjugate to s. 



i 



272-273] GENERAL THEOREMS 343 

In o- let the letter which replaces *> be denoted by a;/; let 



arranged in some 







f-^'l Xo ... Xj{\ 

~\yi y^-.-yJ' 




where y^, 


2/2. •• 


.yn are the letters «i, x.2,.. 


• *X)'^i cil Att, 


order. 








Now 


<T 


^/«i X., ...xn\^fy, 2/2 
W x.^...Xn] Vy/ 2/2' 


...2/J' 


and 

Hence 




_i f^i X^ ... Xn \ 

\Xi X2 ... X^i J 




cr5a"~^ = 


\yi 


y-2 "'yn\fX-^ X^...Xn\ fX^' 

y2"'ynJ\yi 2/2-"2/JUi 


u/2 • • • t*/-) 




/3/1 


2/2 '..yn\(Xi X^' ...Xn\_ 


/aji' iCg' . 




\yi 


y-2 ■■■ynJ\yi y-2 '-ynl 


V2/1' 2/2' • 



2 • • • ^tt \ 

'/...2/J 



2/»i 

Therefore o-scr~^ is the substitution which would be obtained by 
operating on the expression for s with the substitution a. It must 
then be a product of cycles each having the same number of letters 
as the cycles of s ; and is obtained from s by permuting the letters. 
Such substitutions are called similar'. Every substitution similar 
to s is obtained from 5 by a suitable permutation of the letters, 
and is therefore of the form crso-~\ 

Now if s be any cycle {xiX^ ...xjc) then 

S~'^ = (Xk . . . x^x-^) 

as may easily be verified ; and hence s~^ is similar to s. But 
every substitution is a product of a number of independent cycles, 
the inverse substitution is then the product of the inverse cycles ; 
hence any substitution is similar to its inverse. 

273. If m substitutions Sj, 50, ..- Sm are such that the product 
of any two of them is itself one of the m substitutions, these m 
substitutions are said to form a group. 

Thus as may be at once verified 

i, (^^1^2^ 5 1, {XiX2Xg), {X1XSX2) 

are groups. 

The number of substitutions included in a group is called the 
order, the number of letters affected is called the degree of the 
group. 



344 THE ALGEBRA OF INVARIANTS [CH. XVI 

Thus the two groups written down are of order 2 degree 2, 
and of order 3 degree 3 respectively. 

Now the n letters x^, sc.,, ... x^ can be arranged in n\ ways, 
hence the total number of substitutions affecting n letters is n!. 
These substitutions obviously form a group, it is of degree n and 
of order n\. This group is called the symmetric group for the n 

It is useful to have a symbol by which to denote this group, 
the symmetric group for the letters x-^, x.^, ... Xn will be written 

\X1X2 • • • Xn\ . 

More particularly this symbol will be used to denote the sum 
of all the substitutions of the symmetric group. 

Again the product of two even substitutions is obviously an 
even substitution, hence the even substitutions which affect n 
letters form a group. This group is called the alternating group. 
Let 

Sl, S^y • • • Sjfi 

be the members of the alternating group, and 

the remaining substitutions which affect the letters Xi, X2, ... Xn. 

Then these latter substitutions are all odd ; and hence the 
product of any two of them is an even substitution. 

Now if ti, t^, tz be any three substitutions and 

then t~^ ti U = tr^ ti U , 

and hence ^2 = ^3- 

By hypothesis the substitutions Si, s^, ... Sm, (Ti, <t2, ... a-m' a,re 
all different, hence the substitutions 

are all different. But the former set include all the substitutions 
of the n letters, hence the latter must do so too. Hence the even 
substitutions 

form the alternating group. And therefore 



273-274] 



GENERAL THEOREMS 



345 



The symbol {osiX^ ... a;„]' 

will be used to denote the sum of the even substitutions minus 
the sum of the odd substitutions of the letters 

Xif X2, ... Xn • 

This will be called the negative symmetric group, and on 
the other hand 

will be called the positive symmetric group of the n letters. 
For example 

{•^l'^2'^3/ ^^ 1 "t" (•^1^2'^3) "^ C'^l'^3^2) ~ {•^v'^2) ~ ('^2'^3) ~ V^S'^l)' 

274. As an illustration of the notation just introduced we 
remark that the determinant 



hi 62 



a„ 
bn 



may be written • 

[ah ... k}' aib.2 ... kn, 

the substitutions being supposed to affect the letters and not the 
suffixes. Or adopting a double suffix notation we may write 



n, 1 <■% 2 



a, 1 a. 



"2,1 



2, 2 






n 



— [ttiCl^ ... (Inj (ii,iCl2,2 '•• Cin,n> 



Cf'n, 1 (f'n, 2 • • • ^n, ?i 

where the first suffix only appears in the substitutions and is alone 
affected by them. 

Or again 

aP bP cP : 

a9 6? c5 ={a&c}'a^6?c''. 

a*" b"" e 

As an example of the use of the positive symmetric group, 
referring to § 44, we observe that the rth polar of the form 



aa;'* = a, a, ... a,. 

XX X 



346 



THE ALGEBRA OF INVARIANTS 



[CH. XVI 



may be conveniently written 



275. Consider any function F of the coefficients of certain 
linear binary forms a^, /3x, jx, which is homogeneous and linear 
in the coefficients of each form separately. We may write 

where the ^'s are functions of the coefficients of the linear forms 
7-8, ^x, ■■• of the same character as F. 

Then {a/3}'i^=(a/3)[<^.3-</>3]; 

i.e. {a^y F is the product of (a/8) and a function which does not 
contain the coefficients of oCx or /3a;. 

Again we may write 

F=l,Clr^syt<l>r,s,t, {r,S,t= 1,2), 

but here the suffixes r, s, t can never be all different. 

Now 

[a^i\' ci,f^s^t= %■ as at 
A- yS, 0t 

It Is It 

hence {a/37}'i^=0. 

In the same way if Wj , a, , . . . be any ^-ary linear forms, where 

and F be any function homogeneous and linear in the coefficients 
of each, 



{fitiOa ... Op]' F = 



«1, ] '^l, 2 

Os, 1 Ota, 2 



a, , 



i>p 



V^, 



I '^i?, 1 *p, 2 • • • ^, p 

where the substitutions affect the first suffixes only, and -v/r is a 
function of the coefficients of 



^p+\' ^p+i^' 



274-275] 



GENERAL THEOREMS 



347 



The expression | aia.2 ... oip | 

will be used as an abbreviation for the determinant just written 
down. 

Again, ii r>p, 

{a,a, ...or,}'#=0; 

and ii r <p, it is easy to see that 

where Ax is one of the determinants of the matrix 



*1, 1 ■*!, 2 



°h, 1 "2, : 



tto 



^, 1 ^r, 2 • • • ^r, p 

It is unnecessary to suppose that a^-^i, a^_2, ••• Or.p are the 
coefficients of a linear jo-ary form a^ . The facts just established 
are true if F is homogeneous and linear in each of any m sets of 
quantities 



02,1 



tl, 2 



'2, 2 



H,p 



So 



0^, 1 



'■m,p 



there being p quantities in each set ; and a one-to-one corre- 
spondence between the members of any two sets. 

Thus in particular : If F is a function homogeneous and linear 
in the coefficients of each of m binary n-ics 

m being greater than n+1, then 

{oi,a, ... oin+.^'F=0 (i), 

{aitta ••• a„+il'i^=| aiOEj ... Un+i \ . F^ (ii), 

{a^cc^ ... an]'F= \ a^oL, ... a„yS | (iii), 

where the substitutions affect the first suffixes only of the coefficients 
^p,q} I *i«2 ••• (X-n+i is the determinant of n+\ rows and columns 
formed by the coefficients of the ?i + 1 quantics concerned ; 

I ttiOa ... 0^^^ ! 

is the same determinant with quantities /3o. A. ••• ^n replacing 



O^n+i, o> Otji+i, 1> ••• O^n+i, «j 



348 



THE ALGEBRA OF INVARIANTS 



[CH. XVI 



these quantities being homogeneous and linear in the coefficients of 
the quantics represented by ctn+i, ci,i+2, ••• o^; (^nd where F^ is 
homogeneous and linear in the coefficients of the quantics represented 

The first two of the above results are sufficiently clear from 
what has already been said. As regards the last we observe that 

where Ax is one of the determinants of the matrix 



^'l, «!, 1 

0^2, ^,1 



*2, n 



'"n, ^i, 1 



<^,n 



That is Aa may be taken to be the minor of On+i, k in the 
determinant 



and hence 



{ajfls ... OL,^YF= ajOo ... ct,,^ |. 



276. Let the quantics of the last paragraph be represented 
symbolically thus 

(Or, 0, Or, 1, • . . Or, n][«?i, a;^)" = [tt^;*'-*]", (r = 1, 2, . . . m). 

Then the determinant | aia2 ... ««+! I is an invariant, for it may 
be written symbolically 

n (a'^'^a^"^) (r, s = l, 2, ... m; r'^s). 

r, s 

Hence if F is an invariant, then Fy^ is also an invariant. 
Again we can shew that if F is an invariant, then 

(/3o, A. .../3«][^i, a^2f 
is a covariant. For let Ar be the minor of a^+i, r in the determinant 

I ftjaa ... ttn+i |. 

Then since F is an invariant 
is also an invariant. 



275-277] GENERAL THEOREMS 349 

Now we know that Sa„+i, ,.-4^ and 2a,i+i,r( \x-^^~^x^ are 
both invariantive, and hence that the quantities 

Aq, Ai, ... An 

and x^'\ i^\x^'''-'^x.2, ...(jx^^^ 

form two cogredient sets. Therefore, since 2^^ J.^. is an invariant, 
2y8y ( ) x^^~^x^ must also be invariantive. In other words 

{^o,^,,'-'^n\x„X.,Y 

is a covariant. 

If F were a covariant, and contained the variables x^, x^, then 
/8o, /3i, ... ^n would also contain these quantities; in this case the 
form 

(/3o, A, ... ^n\yi,y^T 

is a covariant in two sets of variables. 

277. As examples of the use of the results just established 
we may instance the fundamental identity for binary forms 

{abc]'{ah){cd) = 0; 
that for ternary forms 

[ahcd]' (abc) (def) = 0, 

or {abcdV (ahe) (cdf) = ; 

those for quaternary forms 

{abcde}' (abed) (efgh) = 0, 

[ahcde]' (abfg) (cdhi) (ejkl) = 0, 
and so on. 

Let / be an invariant linear in the coefficients of each of 7i + 1 
binary n-ics ; then with the notation of § 276 we have 

[aiCXa . . . an+i}' / = X i ttitta . . . Cln+i | 

where \ is some constant, possibly zero. 

If n is odd we may take 

I = (a<^' a®)» (a<^' a<'")" • • • (a'"' a<"+i')'*- 

In this case, provided the n + 1 quantics are all linearly inde- 
pendent, 



350 THE ALGEBRA OF INVARIANTS [CH. XVI 

is different from zero, for all terms of / in which two coefficients 
have the same suffixes are destroyed by the operator, and the rest 
are obtained from 

(- 1) 2 ( n - 1 ) a „_i a _ „+i 
\~2^/ ""^ '^'^''"^- 

by interchanging both the first suffixes of pairs of brackets in all 
possible ways keeping the order of the three first suffixes un- 
altered — so that the first suffixes of any one bracket are always 
of the form (2r — 1), (2r) and in this order — and by then inter- 
changing the first suffixes inside individual brackets, each such 
interchange being accompanied by a change of sign. But both 
these operations are effected by 

— since the first only requires even substitutions. Hence 

{ajOs ... a„+i}'/ = X {aiOg ... a,i+i}'oi oa2,na3,iO'4,9i-i ••• a «-!« , , n+i 

= ± \|aia2 ••• Oii+il 
where \ is not zero. Hence when n is odd 

lajOa ... a,i+i| 
is reducible. 

Again, if n = 4, it is easy to see that this invariant is irre- 
ducible. Let us suppose that it be reducible, then 

I aia2a3«4a5 1 = 2/2/3 

where If is an invariant of degree r. 

Now /a must be of the form (a'^^'a'*')*, where r, s are two of the 
numbers 1, 2, 3, 4, 5 ; hence 

and therefore {aia2a3a4a5r | aia2*3«4«s I 

= 5!|aia2a3a4a5| = 0. 

This we know to be untrue in general, hence the hypothesis, 
that the invariant in question is reducible, is false. 



277-279] GENERAL THEOREMS 351 

278. Let s be any substitution of the symmetric group 

{aia.2...an}, 

then {aia.^ ... «.„} s = {ctiaa ••• ««}• 

For {aia.2 . . . an] s contains ?i ! terms, which are all dififerent 
(if a and a are different substitutions, then crs and a's are also 
different), and are all members of the positive symmetric group 

{aitts ... On]. 

Similarly s [aiUo . . . a„} = {aiCi^ . . . «,J. 

Now any purely formal relation between substitutions will 
still hold good if the sign of every transposition be changed ; 
hence 

S [aitto... ««}'= ± {(11(1.2 ... ttnY 

according as s is an even or odd substitution, in particular 

(«! a^) {a^a-i . . . an]' = - {aitta . . . a„}'. 
Again if (ajaa ••• ««} be any positive symmetric group and 

{aitt^bs . . . bm]' 

be a negative symmetric group, the two groups having a pair of 
common letters, then 

(aitts . . . an] {tti a^bs . . . b^]' ^ 

= {aia.2 . . . (In] {(ha^) [- (a^a^) {a^a^ba ... bm]'] 

= - (aja, . . . an] {a^aobs . . . 6,«}' = 0. 

Similarly {aiaa^a . . . b^]' {a^a.^ . . . an] = 0. 

Thus if two symmetric groups, one positive the other negative, 
have a pair of common letters, their product is always zero. 

279. The following purely formal theorem enables us to 
establish various results relating to invariants. 

Let the letters a^, a<., ... an be arranged in any manner in 
horizontal rows, so that each row has its first letter in the same 
vertical column, its second letter in a second vertical column, and 
so on ; and so that no row contains more letters than any row 
above it. 



352 THE ALGEBRA OF INVARIANTS [CH. XVI 

Thus for four letters a^a^a^a^ the five possible kinds of arrangement of 
the tableau would be 



a. 



Then form the substitutional expression 

8 = '2GiGo ... G^;iri'r2' ... r^;' 

such that Gi is the positive symmetric group of the letters of the 
first row, G2 that of the letters of the second row, and so on, Gh 
being that of the letters of the last row ; and that F/ is the 
negative symmetric group of the letters of the first column. 
Fa' that of the letters of the second column, and so on, F^;' being 
that of the letters of the last column (in case a row or column 
contains only one letter, it is understood that the positive or 
negative symmetric group of a single letter is unity). 

Let us suppose that in the tableau considered there are eti 
letters in the first row, ofa i^i the second, and so on ; where owing 
to the conditions laid down 

ai + a2+ ... + «/, = 71 J 

«! < "2 <f: tta ••• <t: ka, «! = ^^ 
let Ta„ 02, ...a ^6 the sum of the n ! expressions S obtained by 
permuting the letters in the tableau in all possible ways, the 
numbers ai, ag, ... a/i of letters in the various rows remaining 
fixed. 

Then ^-^a,, aj, ... «^ ^a,, aj, ... o» = 1 

where the summation extends to all possible values of the numbers 
«!, a.2, ...ah which satisfy the conditions (I); and ^tt,.aj, ... a^ is a 
numerical coefficient which can be uniquely determined. 
For two letters we have 

For three letters we have 

1 = ^-^ 3 + i-' 2,1 + Fir-' 1.1.1 

as can easily be verified. 



279-280 J GENERAL THEOREMS 853 

280. Let the T's be arranged in the order defined by the con- 
vention that Ta „. a comes before Tg, b. b , when the first of 
the differences 

which does not vanish is positive. 

Now if S be one of the n I expressions of which T^^^ ^^^^ ^ is 
the sum, then Ta,, a., ... a^ is obtained from S by permuting the 
letters in all possible ways and taking the sum. Hence if when S 
is expanded as a sura of substitutions, any particular substitution 
s occurs in it, then in T^^^ „„, ... a^ the sum of all the substitutions of 
the symmetric group of the letters a^, Oa, ... «„ similar to s must 
occur. Hence defining t^^^ p^^ ... ^ to be the sum of all those 
substitutions which are formed of k cycles of orders 0i, ^^, ... ^8^ 
respectively, it follows that 

Ta„ a,, ... a, = 2X^„ ^,, ... ,3, ^/3„ P,, ... P, (H), 

where \p^^ p,^^ ... ,3 is a numerical coefficient. 

If cycles of order unity, which are equivalent to the identical 
substitution, be introduced, we may suppose that the suffixes of 
tpu /3„, ... ;8 satisfy the conditions 

The t's are now defined by numbers which obey exactly the 
same conditions as those which define the T's. The number of t's 
must then be equal to the number of T's, let us say equal to M. 
Now the equations (II) may be regarded as a system of linear 
equations expressing the ^'s in terms of the T's. Hence if these 
equations are all independent 

tp„ /3,, ... fi^ = S^a„ a.,, ... a„ -^ a„ a.. ... a,, 

and in particular 

■■■ ~ h, 1, ... 1 ~ ^^a-i, a-i, ... a,j -'a,, a.^, ... a^ • 

If these equations are not all linearly independent, there must 
be a relation of the form 

S5T=0. 

G. & Y. 23 



354 THE ALGEBRA OF INVARIANTS [CH. XVI 

In order to prove the impossibility of such a relation, it will be 
shewn that 

(i) ^ai, «„ ... a, ^^„ p,, ... /s,,, = 

when Ta^^ a,. ... a comes after T^s^, ^s^, ... ^s^,, the T's being arranged in 
the order defined above ; and that 

(ii) n.. <..... a, + 0. 

Let S = FN be one of the n ! expressions of which T^^^ „^^ ... „^ is 
the sum, where P denotes the product of the positive symmetric 
groups, and iV that of the negative symmetric groups. Similarly 
let S' = P'N' be one of the expressions of which T^^^ p^^ ... p^, is the 
sum. If one of the groups of P' contains a pair of letters contained 
in any one group of N then, § 278, 

NP' = 0. 

Consider the tableau by means of which S is formed, «! is the 
number of letters in the top row, it is also the number of columns, 
and consequently it is the number of the positive symmetric 
groups in P. Again a^ is the number of letters in the second 
row, and hence «! — a2 is the number of columns which contain 
one letter only. Similarly o^ — otg is the number of columns 
containing exactly two letters, and in general «» — Of+i is the 
number of columns containing exactly i letters. 

If ^i > «! there are more letters in the first row of the tableau 
for S' than there are columns in the tableau for ;S^. Hence one 
group at least of the product P' contains a pair of letters belonging 
to the same group of N ; and therefore 

KP' = 0. 

In order that NP' may be other than zero, we must then have 
(Si ::^ «!• If this condition be satisfied but 

we see that there are more letters in the first two rows of the 
tableau for S' than can be arranged in the tableau for S with 
the condition that no three occur in the same column. In this 
case some group of N must contain three of the letters of the 



280] GENERAL THEOREMS 355 

first two groups of P', and therefore two of the letters belonging 
to one of these groups. Hence if 

then NP' = 0. 

Again if ^i + ^ii + ^3><^i + a, + ^3, 

some one group of N must contain four of the letters belonging to 
the first three groups of P', and again NP' vanishes. 

Proceeding thus we see that NP' is always zero unless 

We deduce that if the first of the differences 

which is other than zero is negative, then NP' is zero, i.e. if 
Ta„ a.,, ... a, comes after T^,, ^^, ... ^^, then 

NP' = 0', 
and hence 

T... ,. ... „, T,,. ,,, ... ,^ = (SPiY) (2P'N') = 0. 

Next we must prove that 

T T =T- =t 

For this purpose it is only necessary to shew that the 
coefficient of the identical substitution is other than zero. Now 

but every substitution is similar to its reciprocal substitution ; 
hence if s be any substitution contained in ^^„/3j, ... js ,s~^ must also 
be contained in this expression. It follows that in Ta^, a^, ... a^ both 
s and s~^ have the same numerical coefficient. In T^. „ . the 
only products which produce the identical substitution are those 
obtained when a substitution s is taken in the first T, and s~'^ in 
the second. Hence the required coefficient of the identical 
substitution is of the form IX-, which, being the sum of a number 
of positive terms, cannot be zero. 

Let us now suppose that there is a relation of the form 

23—2 



356 THE ALGEBRA OF IXVARIANTS [CH. XVI 

and let \T„ „ „ be the first term of this relation, the terms 
being arranged in the order defined above. Then 

[2£r]r„..„^;...„, = o. 

And hence by the relation (i) 

^^\, a.,, ... a,, = ; 

therefore by (ii) X = 0. 

Thus no T can be the first term in such a relation ; in other 
words the equations (II) are linearly independent. Hence 

l = S^„,..,,...a,^a.......a, (HI). 

Q.E.D. 

281. The coefficients in this series have been calculated* but 
as their values are not of importance for our present purpose we 
merely quote the formula 

• n (a,, — ttg — r + sy 



/ n {a,.- as- r + s)\' 
U,, a,. ... a, = yuior + h-ry.) 



The substitutions SiSz and So^i are similar, for 

S1S2 ^ ^2 \^2^1/ ^2> 

their coefficients in the expansion of T^^^ „..„ ... a,, are therefore the 

same. Hence since 

T = 1PN 

it follows that also 

If r^^^pj ... ^^, = ^P'N' comes before Ta„ «, ... a^ it has been shewn 

that 

NP' = 0. 



Hence also 


P'i\^=0, 


and 


N'FNP = 0, 


and 


\%N'F][1NP] = {), 


and therefore 


^/Si. iSj. ... ^v -^tti. o-i, - «A — ^- 



That is, the product of any two different T's is zero. 

* Young, Proc. Lond. Math. Soc. vol. xxxiv, p. 361. 



280-282] GENERAL THEOREMS 357 

Now multiply the relation (III) by T^^, a^, ... a,^, \ye obtain at 
once 

T —A 7*2 

282. By polarizing the form 
once with respect to each of the sets of variables 

r,' (1) 7- (1) r (1) 

«i<2' a;2<2> ... a;,,,<2) 



(m) ^ (m) ^ (m) 



rp [in rf> 

tvo • • . ■^91 

we obtain an expression which may be written 

= — {aWa<2» ... a<'"'}a'iVi'a'"a;<2) ... a"«>a;(m). 

If in F^ each of the sets of variables is replaced by the 
original set x^, x^, ... Xn, we obtain the form F from which we 
started. Neither the passage from F to F^ nor that from F^ to F 
affects the invariant properties of F, so that we may regard F and 
^1 as equivalent. 

Similarly if /(a'^', a'^*, ... a""') be an invariant linear in the 
coefficients of each of m quantics of the same order, whose 
coefficients are 

ao^ ai'^ ... (r = l, 2, ...m), 

then [a^^^a^-^ ... a<'«>}/(a'i', a'^*, ... a<'">) 

is an invariant which may be obtained by means of Aronhold 
operators from an invariant of a single quantic. 

Again if P be the product of li positive symmetric groups, 
which between them contain all the letters a'^*, a'-*, ... a*'*", but no 
two contain the same letter, then 

P/(a"', a^-'\ ... a**")) 

is an invariant which may be obtained by means of Aronhold 
operators from an invariant of h quantics. 



358 THE ALGEBRA OF INVARIANTS [CH. XVI 

283. Peano's Theorem. Let F{a^^\ a^-\ ... a'"") be a 
covariant linear in the coefficients of each of m binary 7i-ics 

(ao"^, a,**-*, ... aj^'^ccu x^f; r= 1, 2, ... m. 

Operate on F with the two sides of the identity § 280 (lii.) 

1 _ VJ T ' 

J. — ^-a.^.^, a.2, ... ttj -^ o., o.,, ... a^) 

then i^=S^.„^^,...„^T,,.^^,._^i^. 

Now T„„^,....^ = 2Pi^, 

where N has a factor of the form 

{a<«a(--=' ... a<^']'. 

If A > 11 + 1, then by § 275, 

[a^'^a^'^'' ... a(*'}'i'=0, 
if A = ri + 1, 

{a^a® ... ff,(A>}'2?^=| aWa<-' ... a<*) \.R, 
if A = 71, 

{a<"a® ... a<'^'}'i?'=| a»'a<=« ... a^^^Q \, 

where {Qq, Qi, ... Qn^Xi,x.^"^ is a covariant of the forms considered. 

Hence when h>n-\-\, 

Ta„a„...aF=0; 

when h = n + l, Ta^,a.„ ...a^F is a sum of terms each of which 
contains a factor of the form | a'^'a'^' ...a'"+^' j; when h = n, 
Tai, a^, ... a^^ 18 a sum of terms of the form j a^^^a^-^ ... a'"'Q |. 

Now the number of positive symmetric groups in P is h; 
hence by § 282, Ta^,a.„...a^F is a sum of terms each of which is 
obtainable by means of Aronhold operators from a covariant of 
only h different 7i-ics, 

Let us suppose that P is a covariant type which does not give 
any irreducible covariant, unless we are considering a system of 
more than n w-ics. Then 

if h<n-\- 1, Ta„ 02, ... a.^F is reducible for it is obtained by Aronhold 
operators from reducible forms : if /< > n + 1 

^ai.cHj, ... o*P= 0; 



283-284] GENERAL THEOREMS 359 

and if k = n + l, Ta,,a,,...a^F is a sum of terms each of which 
contains a factor of the form | a'''^''a^^'' ... a*""^^' |, and is thus 
reducible unless m = w 4- 1. 

Hence evei'y type of the binary n-ic which does not furnish 
an irreducible covariant for a system of n n-ics is reducible, 
with the possible exception of the invariant type 

I a^^^a^^^ ... a"^+i» |. 

Further it has been shewn that, if h = n, Ta^, a.^, ... ai^F is equal 
to a sum of terms of the form i a^^^a^^^ ...a^^^Q . But if the 
invariant | a^^^a^-^ ... a"'"*"'* | is reducible, the invariant 

is reducible in the same way. Hence when the above invariant 
is reducible every irreducible type furnishes an irreducible co- 
variant for a system of n — 1 w-ics. 

It has been shewn § 277 that this invariant is reducible when 
n is odd ; thus the covariants of any number of cubics can be 
obtained by means of Aronhold operators from those for two 
cubics. 

Similar results may be obtained in exactly the same way for 
ternary forms, or for forms involving any number of variables. 

Thus all covariants of any number of ternary ?i-ics may be 

obtained by means of Aronhold operators from the system for 

n + 2 



, — 1 ternary vi-ics with the possible exception of the 

()i + 2\ 
j rows and columns, each row 

being formed by the coefficients of one n-ic. 

Ex. Shew that the determinant formed by the coefficients of six conies is 
an irreducible invariant of the system. 

284. Peano's theorem was first proved l)y means of an ex- 
pansion due to Capelli*, which is virtually the same as the 
expansion used here, but is expressed in terms of polar operators. 

* " Sur les Operations dans la Theorie des formes Algebriques," Math. Ann. 
Bd. 37, See also "Lezioni sulla Teoria delle Forme Algebriche," Ch. I. § xxiii. 
by the same writer. 



360 



THE ALGEBRA OF INVARIANTS 



[CH. XVI 

It is sufficient, in order to make the comparison clear, to point 
out that if / be a function linear in each of n sets of ^-ary 
variables, a^, a^, ... an', then 

where A^ is one of the determinants of the matrix 

^,1 '^i.a ^,9 

Ctg, 1 C^S, 2 '^a 



«'2,5 



and in fact is equal to Ha„a^,...a^f where 
Dx being the determinant of the matrix 





d 
9ai,2 '" 


d 

'" d(li,q 


d 


d 
Bag, 2 


d 

'" da2,q 


d 


d 


d 



dahi caji 



dah,c 



obtained by writing ^ — for ar,s in each element of A\. 

(Jdr, 8 

In the case of binary forms, if a^^h^^ be polarized so as to 
obtain a function of m + n sets of binary variables, and then the 
identity § 280 (lll.) be applied, we obtain a proof of Gordan's series, 
§ 52, which is a particular case of the series of Capelli. The 
coefficients in Gordan's series are not apparent, but it is possible 
to obtain them by these methods*. 

Thus if /= aa;(i)aa;'2* . • • <*«'">> &j/'i' ^2/*2) . . • hy(n) , 

then T.„a,,...aj=0, 

when h> 2. 

If h = 2, we need only consider those expressions PN which 
are of the form 

PJV= 

{a;Wa;<2> ...a;""'y'^* •••2/'*'l {y**+'*?/''+^' •••2/'"*} {x^^^y^^'^^'Y ... {aj"'-^'^/"*'}'; 
* See Young, Proc. Lond. Math. Soc. vol. xxxiii. 



284-285] GENERAL THEOREMS 361 

and ill this case 

PJSf.f=P.(x^'^y^'+'^) ... (x^"-^h/''^)(ahy'-'a^(n-i+i) ... aa;(m)6y(i) ... byU) , 

which may be obtained by polarization from 

(xyy-^ {oh)"--^ a^^^-'^ 6/ 

In just the same way we see that, if x, y, z be ternary variables, 
ax"^by^CzP can be expanded in a series each term of which may be 
obtained by polarization from a term of the form 

X {xyzf (abcY (abxyY^ (bcxyy^ (caxyy^ a^^ bx^Cx^, 



where 



{abxy) = 



Let 



then 



tti Oj x^yz x^y^ 
a^ O2 x^y-i — Xiy^ 
(h bs x^y^-x^^yj 
III ^=^23/3 ^33/2 
^2 = ^3 2/1 ^1^/3 
W3 = ^1^/2 ~ ^22/1 ) 
u, = 



is the condition that the point z lies on the straight line joining 
the points x and y. Hence Uj, a.^, u^ are really the coordinates 
of a straight line. 

We see then that all covariants of ternary forms which contain 
any number of variables can be expressed in terms of polars of 
covariants which contain the variables x and u only. The m's were 
introduced for geometrical reasons § 207, but it is now apparent 
that they are necessary to make the analytical theory complete, 

285. Let a;'^', a;*^*, ... a;*'*' represent n sets of q-&vy variables 

^i**"', ^•o*''*, . . . ^9*'"' ; r = 1, 2, ... /I. 

Then if i'' be a function linear in each of these sets, we see, in 
the same way as before, that PNF is a function obtainable by 
polarization from a function, not necessarily linear, of 



1st the sing-le set 



7' (1) 






2nd the determinants of the matrix 



X.2 



Xq^^ 



362 THE ALGEBRA OF INVARIANTS 

3rd the determinants of the matrix 



[CH. XVI 



'3) 






g,<a, ^ is) 

and 80 on, and lastly of the determinant 



«. (2) 



J- (1) 

tlyq 



.w 



'5' 



a^5*«' 



These may be all regarded as auxiliary variables. It will be useful 
to denote the variables of the 2nd set, viz. the determinants of the 
matrix 



r (1) 



by the letter 2«, w'ith appropriate suffixes. Similarly the third set 
will be denoted by gX and so on. 

Then in taking the complete system of concomitants of 5-ary 
forms we have q — l different kinds of variables which may 
appear. 

The geometric meaning of these variables is easy to obtain. 
A space of g- — 1 dimensions being under consideration, the variables 
cc or iX represent point coordinates : the variables ^ are line 
coordinates; the variables 3X are plane coordinates, and so on. 

The linear substitutions by which these auxiliary variables are 
transformed, when any linear transformation of the point coor- 
dinates is made, are easy to find. The variables {oc and q_iX are 
contragredient, and in fact 

•i* {X q — {X 

is an absolute concomitant. As a particular case, when q is even *, 

this leads to a relation between the variables qX. This remark 

2 
has already been illustrated for quaternary forms § 220. 



The case of binary forms is of course an exception. 



285-286] GENERAL THEOREMS 863 

286. Let ax^^\ «a;'-*, ... a^^"'^ be any 7i linear q-ary forms; and 
F a function linear in the coefficients of each. Then apply the 
formula 

F=2A T F 

where substitutions interchange the sets of coefficients 

aW, a<2), ... a'«'. 

Consider the form PNF, it is a sum of terms each of which 
is a product of determinants of matrices of the form 



ttj 


(1) 


a. 


(11 


...a^^' 


a, 
a. 


(2) 


a^ 


(2) __ 


...a,<^'' 


(t) 


a-. 


CM ._ 


...agC-" 



Moreover if iV" has two negative symmetric groups of degree i 
and Ai^x, Si,fi represent determinants from each of the correspond- 
ing matrices (the particular determinant being defined by the 
second suffix), then every term of NF has a factor of the form 

Hence every term of PNF has a factor of the form 

Ai^K-Bift + Ai^fiJji^\ 

as is evident when the tableau from which PN is constructed is 
considered. But this is a coefficient of the expression 

[tAiix][l.Biix] 

which is a covariant since the as are contragredient to the x's. 

Now if PN is a term of 2"^^. „.„... a,., N" contains a^ groups of 
degree h, Uh-i — cch groups of degree h — 1, and so on ; hence PNF 
is a linear function of the coefficients of concomitants of the form 

Thus every function of the coefficients of certain linear forms 
can be expressed in terms of coefficients of concomitants of those 
forms. 

It is unnecessary to assume that the function is linear in the 
coefficients of each of the forms, in order that the above theorem 
may be true. For the function can be made linear by means of 



364 THE ALGEBRA OF INVARIANTS [CH, XVI 

Aronhold operators, and after the above process the original 
coefficients can be restored without affecting the invariantive 
properties in question. 

Further the forms considered may be symbolical, and we at 
once deduce that every integral function homogeneous in the 
coefficients of each of certain q-ary forms can he expressed as a 
linear fuTiction of the coefficients of the comomitants of those 
forms. 



APPENDIX I. 



NOTE ON THE SYMBOLICAL NOTATION. 

As we have said in § 82 the notation used in this work 
is really equivalent to Cayley's hyperdeterminants. The great 
advance made by the German school lies in the possibility of 
transforming symbolical expressions, and, of course, in the proof 
that every invariant form can be represented as a combination 
of hyperdeterminants. The reader may feel the need of justifying 
directly the results obtained by manipulating umbral expressions 
and accordingly we shall indicate how the whole theory can be 
made to rest on differential operators. 

There are different ways of doing this. Salmon has remarked 
that, /being a binary form of order n, since 

we may regard/ as being equal to 

where "^"'''W.) [fyj ' 

Hence we may suppose that 

5 d 

and that the final operation is on 

A 

nl' 



366 THE ALGEBRA OF INVARIANTS [aPP. I 

Any symbolical expression can be thus at once transformed 
into one exactly like it but involving only diflferential operators, 
e.g. if 

then (a^y aaT-' ^x'''' 

is ( _s^ ?Lyf^ 1+^ ir~7^ 1+^,1^^ fe 

\dy1dZ2 dzjdyj V ^dyi ^dyz) \ ^dz-t, 'dzj m\n\ 

(see § 82). 

In any calculation we may omit the operand while we trans- 
form the operator. 

After this the reader who wishes to do so will have no difficulty 
in developing the theory of the symbols when they are regarded 
as differential operators. 

For another method see Kempe, Proc. L.M.S. vol. xxiv. p. 102 
and Elliott, Proc. L.M.S. vol. xxxiii. p. 231. 

Some interesting general remarks on the underlying principles 
of the symbolical notation will be found in Study, Methoden 
Terndre Formen ; and some very curious remarks in Lie-Scheffers, 
Vorlesungen Uber Gontinuier lichen Grilppen, p. 720. 

This is the most convenient place to give a brief explanation 
of the so-called Chemico- Algebraic theory — an idea originally due 
to Sylvester which has perhaps attracted more attention than its 
intrinsic merits deserve. 

In this theory an atom in chemistry corresponds to a binary 
form in algebra, and the valency of the atom to the order of the 
form. To each unit in the valency of an atom, in the chemical 
theory, a bond is supposed to correspond, and each such bond can 
connect the atom in question with an atom of valency one such as 
Hydrogen. Thus Oxygen is of valency two, and there exists a 
compound OH2 which is written graphically 

H— 0-H, 

there being two bonds proceeding from O and one from each H. 

Since each unity in the order of a form gives rise to one 
possibility of trans vection with another form, the analogy is evident 



APP. l] NOTE ON THE SYMBOLICAL NOTATION 367 

— if we have a binary quadratic o^^ and two linear forms Aa;, ^x tbe 
formula OHg corresponds to the algebraic expression 

iph) {oh') 
an invariant of the three forms. 

Then Carbon being of valency four we have the compound 
(Marsh Gas) 

H 

I 
CH4 or H— C— H 

I 
H 

and this corresponds to the invariant 

{ch){ch'){ch"){ch"') 

of a binary quartic and four linear forms. 

The four hydrogen atoms in CH4 are supposed on chemical 
grounds to occupy similar positions* in the structure of the 
compound and hence CH4 is more naturally like 

{chy 

an invariant of a quartic and a single linear form hx. 

Then the compounds CH3CI, CHoClo etc. may be supposed like 
the invariants 

(chyick), (chy{cky 

where k is written for CI and we see in the chemistry an analogue 
to polarizing in algebra. 

Guided by the above the reader will have no difficulty in 
writing down an invariant corresponding to any graphical formula 
however complicated — in fact the algebraic form of the invariant 
is only a different (perhaps a more concise) way of writing down 
the chemical formula. 

Difficulties arise when we recollect that some atoms have 
apparently different valencies illustrated by S in SO., and H2S, 
for of course a binary form can have only one order. Gordan 
and Alexeleff suppose that the corresponding algebraic form is 
then polarized. Thus S in SOo would correspond to Sx'^ and S in 
HoS to Sx^Sy- and now the degree available for transvection is two. 

* For an explanation of this and the other chemical facts we have referred to 
see Scott, Chemical Theory, chap. vi. 



368 THE ALGEBRA OF INVARIANTS [aPP. I 

The idea has been developed by various writers in the 
direction of making the algebraic methods graphical (references 
in Meyer) and lately Gordan and Alexeleff * have written several 
papers in which the algebra is applied to chemistry ; the papers 
were criticised by Study f and from the objections and the replies 
the reader may be able to form his own opinion. We venture 
only to remark that the wonderful feature of the algebra is the 
capacity for reduction, and that, unless there is something cor- 
responding in chemistry, the whole theory seems to be no more 
than a superficial analogy. It is of course certain that a reducible 
invariant often corresponds to a stable compound — moreover the 
general features which lead an algebraist to suppose a form 
reducible and a chemist to suppose a compound unstable are as 
nearly opposite in character as they can be. 

It has been stated in Chapter i. § 21 that there are two 
ways of obtaining relations between concomitants symbolically 
expressed, viz. : 

(i) By means of the fundamental identities 

{he) ax + {ca) b^ + (ab) Cx = 0, 

(6c) {ad) + {ca) {bd) + {ab) {cd) = ; 

(ii) By means of the fact that a concomitant is left unaltered 
when a pair of letters which refer to the same quantic is inter- 
changed. 

We give here a demonstration of the fact that all relations may 
be thus obtained :|:. 

Let F{Gi) = be any identical relation (supposed rational 
integral and homogeneous) between concomitants Ci of any system 
of binary forms. 

Let F{Ci) = 'ZPj 

each term Pj being itself a concomitant. Also let 

(^0,^1, ... ^nfc.^2)" = a^" = 6=r"=-.. 
be one of the quantics of the system. The introduction of the 
symbolical notation may be effected by operators like 

"■"ax/ ''■""■''=ai. + ••• + "•" ax 

* Wiedemann's Annalen der Physik, 1899, 1900. t In Wiedemann. 

X Cf. Gordan, Invariantentheorie, Bd. ii. § 117. 






APR l] NOTE ON THE SYMBOLICAL NOTATION 369 

operating on F{Ci). These operators do not destroy the property 
that F{Ci) considered as a function of the coefficients of the 
quautics is identically zero. 

Hence if F{Ci) =XPj becomes "StUk, where each term Hk is a 
symbolical product of factors of the forms (ab), Ux, 

considered as a function of the symbolical letters. 

Owing to the manner in which the symbolical letters were 
introduced no distinction was possible between two letters which 
refer to the same quantic, hence XHk is unaltered by any inter- 
change of two such letters, and therefore the second method 
of obtaining relations between forms will have no effect here. 

Let Oi, «2, Us, ... Ur be the symbolical letters, and n^.n^, ... rir 
the orders of the quantics to which they refer — if the forms Ilife 
are covariants we shall suppose that a,-,2 = *'i, ctr,i = — ^2 ^-nd that 
71,. is the order of 11^. Then applying the theorem of §47, x being 
replaced by «i and 1/ by ctj, we obtain 

d 

where SIIj;*-" does not contain da- 

Now SUk = 0, hence every term of this series is identically 
zero. For if the jth term be the first which does not vanish, we 
may divide by {aicuy, and then put aa = di, whence 

\;[Sni<^'] = o. 

But this series is merely obtained by repeated use of the 
fundamental identities (see § 46), hence the reductions used so far 
belong entirely to the two classes mentioned. 

It may happen that '^TLk^^^ is not identically zero considered 
as a function of symbolical factors. In this case we may\^ipply the 
same process again. 

Each time we do this the number of letters in the function 
under consideration is reduced. Hence in (r — 2) steps at most 
the expressions are reduced to expressions which are identically 
zero when considered as functions of the symbolical factors — for 
when only two letters are left there is only one possible symbolical 
factor. 

Thus the identity F{Ci) = is made to depend entirely on 
the two fundamental methods of reduction. 

G. & Y. 24 



^ "' • r 

2,11^ = "S^Xjioiasy a^ 
i=o L 



APPENDIX 11. 

ON WRONSKI'S THEOREM AND THE APPLICATION OF 
TRANSVECTANTS TO DIFFERENTIAL EQUATIONS. 

The form of Wronski's theorem used in § 189 is not the usual 
one, but the determinant, which there vanishes, can be transformed 
in the following manner. 

If f^aj\ 

then 5-^^^ — = 7 ^ aa;"-^-'^ a^^a.f, 

and 

__ I 



(w — A, — /i-) ! 
It follows that 

where the ^^'s are numbers depending on n, r, X and fju. 
Now in § 189 the determinant whose typical row is 

dx^"-' dxi'-'dx^'"' dx^"-' 
vanishes, hence so also does that whose typical row is 

dx/' '"'dxr^dx,''"''' dx/' 



APP. Il] WRONSKl'S THEOREM S71 

On using the value found above for 



the typical row of the determinant becomes 

and since this determinant vanishes we infer, on modification 
of the columns, that the determinant whose typical row is 

dec/' dx/-^'"'-^ 
vanishes. 

If we replace x.^ by unity in the /'s, this is the usual form 
of Wron ski's determinant. 

For the sake of completeness, we shall give an easy proof 
of Wronski's theorem. 

If Ml, M2. •■• Un+i be 71+ 1 functions of a single variable so, and 
the determinant whose rth row is 

**'■' 'd^' dc^""dc^ 
vanish, then there is an identical relation of the form 

XiMi + \Mo + . . . + X„+i lln+i = 0, 

where the X,'s are constants. 

In fact the vanishing of the determinant is the condition that 
the w's should be solutions of the same linear differential equation 
of order n, say 

We have therefore to prove that such an equation cannot 
have {n + 1) linearly independent integrals. 

The theorem is easy to establish when n is unity, so we 
assume it true for n — 1 and proceed inductively. 

Now «i being a solution of the equation of order n, write 

y = Ui fivdx. 

24—2 



372 THE ALGEBRA OF INVARIANTS [aPP. II 

It is quickly seen that w is given by an equation of order 
n — 1, say 

and this equation is satisfied by 

d^ fu^ d I'uA d^ fUn+A 
dx \Va) ' doc \uj ' '" dx\ Ui J ' 

hence by hypothesis there is a relation of the type 

Integrating and multiplying up by u^ we obtain a relation 
of the form 

between the (n+ 1) functions u, so Wronski's theorem is completely 
established. 

The application of the result to the (r + 1) functions / shews 
that if x^ be replaced by unity, there is a relation of the type 

\fi + ^2/2 + • • • + Xr+ijr+i = 0, 

and then making each / homogeneous again by the introduction 
of flJa, the result quoted in § 180 follows at once. 

The device of changing from differential coefficients with 
respect to two variables to differentiation with respect to a single 
variable is often useful. 

For example, the rth transvectant of 

is Vr = (a6)'-a^'"-'-6a,'*-*'. 

Thus x^^yfr = {a-Jb^^ — aaftiiCsy^*"'"'"^*""'" 

^ {n-r) \ dj ^ {n-r + \) \ (m-1)! d'-'f 80 
~ n ! dx{ ^ n\ ml dx/~'^ dxi 

m ! ox/ 



APP. Il] WRONSKI'S THEOREM 373 

Applying this to the nth transvectant of two forms / and (f), 
each of order w, we can reduce the problem of finding a form / 
apolar to <f> to the solution of a linear differential equation of 
order n ; it follows at once that there are not more than n linearly 
independent forms, and moreover from the algebraic theory we 
infer that all integrals of the equation are polynomials in Xi. 

Conversely a differential equation whose coefficients are poly- 
nomials can be reduced to a relation between transvectants. To 
give a simple example, consider the equation 






daf- dx 

the P's being polynomials in x. 

If ^1, 02 > <\>i be three forms of orders r^, rj, r^ respectively, 
and / be a form of order n, 

' v/^p^r - ,, (^ _ 1) <P3 g^^2 ^,,^ 5^^ 9^^ + ^^ (^r, -ly dx,^ • 

Now replacing x.2 by unity as usual, we can choose <^i, ^2,^3, 
so that 

is the same as 

for the transvectant relation is 

1 . ^f _l_dl #3 ^ 1 .rf^0, 

n (w — 1) ''^^ cifa?!^ m-g cla;i c?a?i ^3 (rg — 1) •' c^a?!^ 



and we must have 



\ , df 1 ^d<f>. ., „ 



03 = -Po, 



rj (w — 1) 



1 2 rf03 _ p 

?2 ^ nrg dxi 

1 6^02 1 (^'03 ^ p 



374 THE ALGEBRA OF INVARIANTS [APP. II 

The first equation gives <Jjs and its order is that of Pq, the next 
equation gives (f>2 and then the last gives ^i ; hence the trans- 
formation is always possible, but of course the coefficients depend 
on the order of the form / that is chosen to represent y. 

References to further developments in connection with Differ- 
ential Equations will be found in Meyer's Berichte and in Klein's 
lithographed lectures on Linear Differential Equations of the 
Second Order. 



APPENDIX III. 

JORDAN'S LEMMA. 

In Chapter iv. gi-eat use was made of this theorem: — If 
x + y + z = 0, then any product of powers of x, y, z of order n 
can he expressed linearly in terms of such products as contain one 

exponent equal to or gi-eater than -^. 

In § C4 a proof due to Stroh is given. We shall now give 
a simpler proof iu which a much more general theorem is 
incidentally established. 

The general theorem may be stated as follows : — 

It ax, Ox, Cx, ... 

be a system of r distinct linear forms and 

a, /3, 7,... 

be r positive integers satisfying the relation 

a + + y + ... = n-r + l, 

then it is impossible to find binary forms 

A, B, C, ... 

of orders a, /3, 7, . . . 

respectively such that 

ax"-' A + hx"-^B + Cx''-yG + ... = (I). 

In fact, suppose that such an identical relation exists and 
operate a + 1 times with 

a d J. 



376 THE ALGEBRA OF INVARIANTS [APP. Ill 

It is easily seen, by making a.^ zero, that i)"+^ annihilates a 
binary form of order n only when that form contains the factor 
tta;"""; hence since a^, bx, Ca;, ... are all different, we have 

l^n-p-a-i B' + c^'^-y-a-i C" + . . . = (II), 

where B', G',... 

are of orders /3, 7, . . . 

respectively and do not vanish identically unless 

B, C, ... 
do also. 

The relation (II) is of the same form as (I) except that r is 
changed into r — 1 and n is changed into ?i — a - 1 for 

+ y+ ... = n-r-a + l=(n-a-l)-{r-l)+l. 

Now a relation of the type (I) is impossible when r = 1 for any 
value of n unless the form A (the only one occurring) vanishes 
identically, hence by induction it is impossible for all values of 71 
and the theorem is established. 

The forms A, B, C,... 

involve a + 1, y8 + l, 7+I, ... 

arbitrary coefficients respectively, or in all 

a + ^ + y+ ... + r = {n + l). 

Any binary form can be expressed linearly in terms of ()i 4- 1) 
linearly independent forms of the same order, and since there is no 
identical relation of the form 

aa,"-« A + 6a;"~^ B + Cx''-y C + . . . = 0, 

it follows that any binary form of order n can be expressed 
uniquely in the foi-m 

a^"-«^ + 6it"~^ B + Ca^-y + ... , 

where a^, bx, Cx, ■-• are all different. A, B, G,... are of orders 
a, y9, <y, ... respectively, and 

a + y8 + 74-...=w-r+l. 

Consider now three linear forms 

^, y, z, 

where z = — {x-\-'y) and a;, y are the variables. 



APP. Ill] Jordan's lemma 377 

It follows from the above that if 

a + i3 + y = n-2 

then any homogeneous expression of x, y, z of order n can be 
expressed in the form 

a;"-" P + ^"-^ Q + ^»-y R, 

where P, Q, U are of order a, /3, 7 respectively. 

In other words, changing a into n — X, into ?2 — /i and 7 into 
71 — V, any homogeneous product of order n of cc, y, z can be 
expressed in the form 

.'/^ P + 2/'" Q + z" R, 

where X + fx, + v = 2n + 2, 

and the expression is unique. 

This is Stroh's generalized form of Jordan's lemma given in 
§ 64, and the lemma itself follows at once since we can always 
choose integers X, fx, v satisfying the relation 

\ + /A + i/ = 2n + 2 

, 2n 2n 2n 

and ^^3' ^^"3' ^^"3* 

On expressing P in terms of x and y, Q in terms of y and z, 
and R in terms of z and a?, it follows that any homogeneous 
product of order n of x, y, z (a; + 2/ + 2 = 0) can be expressed 
linearly in terms of 

a?", x'^~^y, x'^~'-y-, ... x^y^^~^ 

yn^ yn-iz, y'''-"-Z-, ... y^^z"'-*^ 

2", 0''-i a^, 2»-- a;^ ... z" x''-" 

and the same would still be true if z were changed into x in the 
second row. 

The reader will have no difficulty in modifying tlie above so as 
to obtain another proof of the fact that the general system of 
apolar forms constructed in § 178 contains n linearly independent 
forms. 



APPENDIX IV. 

FURTHER RESULTS ON CO VARIANT TYPES. 

The expression given in § 262 for perpetuant types, may be used 
to determine the perpetuants when the forms are supposed not all 
diflferent. 

The result (using the notation of the paragraph referred to) 
for one quantic is 

^-8-1 = 1 + ^S-i 
\s-2= 2 + ^5-1 + ^5-2 



X, = 2^-^ + 2(|5-: + rfi-2+...+^2+^0> 

where all the ^'s are zero or positive integers. 

For two quantics the covariant can be written in the form 

((ha^)^^ (a^asY^ . . . (tti_ia,-)"»-i (aAYiib^b^Y^ (hobsf^ . . . (bj^.bjfj-i, 

where the a's refer to one quantic and the b's to the other ; the 
indices satisfy the conditions 



2a, = 2i+i-^ + 2 (^,_i + ^,_+...+ 10. 

For the proof of these results see Grace, " On Perpetuants/ 
Froc. London Math. Soc, vol. xxxv., p. 219. 



APP. IV] TYPES 379 

The reasoning of this paper for the case of a single quantic 
may be applied to types, with the result that all perpetuant types 
of degree B may be expressed in terms of products of types and of 
types of the form 

(ai a.f^ (aoUs)'^^ . . . {as-i asYs - 1 , 

where the X's satisfy the conditions laid down above for types for 
a single quantic, except that 

and where the order of the letters ctj, a^, ... is no longer fixed. 

The method of §§ 262, 265 can be extended to the case of any 
concomitant which is linear in the coefficients of each of certain 
binary forms 

Oi '^1, a, '^s ... 0,5 '^5. 

Thus any such concomitant can be expressed by repeated 
transvectiou in terms of 

(i) forms of the type 

(( ... {{{a^a.^^m^YmiY^... a^-O^s -2 05)^5-1, 
(ii) reducible forms. 
The relations satisfied by the indices are however somewhat 
complicated. They are given as follows : — 

A reduction (in the sense of § 265) is always possible when 
\i < 2*~^ unless one of the following conditions is satisfied : 

\ + S (2X; + 2^-^- - n,-+0 > n, - (2«-'- - 1 ), 
X, + S (2Xj + -I'-J - nj+,) > n, - (2«-^ - 1 ), 

where B> i >j > I, but i and j are otherwise unrestricted, as also 
is the number of terms under the sign of summation — in particular 
there may be none. 

In general, the condition that Xj = 2*~- — a may not mean a 
reduction is that certain positive integers, 

/i(0),/.(0),/;(0); 

/i(|i),/.(liX/3(^i). ^1=1,2; 

/i(|i, ^2\M^i, ^.),/s(^i, ^.), |i=l. 2: f, = l, 2; 

M^^, I., - Ir), /.(I:, I2, ... U /adi, I2. ... ^), 

^,= 1, 2: |, = 1, 2:...^r=l, % 



380 THE ALGEBRA OF INVARIANTS [aPP. IV 

can be found such that 

a =/ (0) +/.(0) +/3(0) +/(1) +/(2) 

m) =/i (r) +/. (^) +/3 (?) +/(!, 1) +/(?, 2) 



/(?!, f., ... L) = A(|., f., ... ^)+/.(|:, |„ ... I.) 

+/3(|l, ?2, .... |r)+/(fl, I., ... Ir, l)+/(|a, |„ ... ^r, 2); 

(where the quantities /(fi, ?2) ... fr) are positive integers which 
satisfy 

/(?:. h, ... ^, l.+O > 2^-^- -/,(?„ ?„ ... ?.)-/3(?.. |„ ... ^) 

when ?i + ?2 + . . . + |r + f r+i + r is odd, and 

/(?!, ^.. ... I.. ir+x)>2«^- -/,(!„ r., ... ^•)-/3(ri. r.> -. ^) 

when ?i + ?2 + • . • + |r + ?r+i + »" is even) ; 

V+2 + 2«-'-^ + </>l(?i, ?2, ... rr)>«i+/i(?n ?2, ...?r)-l*, 
V+2 + 2«— ^ + «^2(|l, ?2, ... |r)>«2+/2(?x, I2, ... ?r)-l, 

X,+, + 2«-'-^-/(?„ |„ ... ?.)>rj,+3+/3(|i, I2, ... ?r)-l ; 
where, finally, the ^'s are defined by the laws 

<A,(ii. r.. - r.) +/(?:. ?., .- ?r)-</>jfi, ?2, ... r.-i) 

is zero if ?/ + ?i + ?„ + • • • + l^r is even, but if this sum is odd, the 
expression considered 

= (2V+i + 2«—-7l,^,)-2{y(f , ?„ ... ?.-:)-/(?>. I2. ... ir% 

and <^,(0) = -a = -/(0), 

<^, (1) +/(!) = /.,- 2 {/(O) -/(I)}, 

(^2(1)+/(1) = 0, 

<^,(2)+/(2) = 0, 

02 (2) +/(2) = /^. - 2 {/(O) -/(2)1. 

* In case /i(fi, lai ••• fr) is zero, this inequality need not be satisfied; this 
remark applies also to the other two inequalities. 



INDEX. 



Alexeleff 367, 368 
Algebraically complete system 
for binary forms 150 
for ternary forms 265 
Anharmonic ratio (see Cross-ratio) 
Apolarity of binary forms 
defined 213 

the a linearly independent forms of 

order n apolar to a binary n-ic 214 

there are s linearly independent forms 

of order n apolar to ii-s + l given 

binary ;/-ics 220 

the cubic apolar to three given cubics 

220 
forms of different orders 225 
the 71 linearly independent forms of 
order m apolar to a binary in-ic 
(tKni) 226 
the forms of order n apolar to a given 

7K-ic {n<m) 228 
application to canonical forms 229 
Apolarity of ternary forms 
of two conies 299 
of cm'ves in general 303 
Curves which possess an apolar conic 

304 
Cubics which possess an apolar conic 

312 
Quartics which possess an apolar conic 

313 
Conies apolar to fundamental conic 
308, 310 
Argand diagram, 

Interpretation of invariant properties 
of binary forms by means of the 
208 
Aronhold 70, 84, 357 
Aronhold operator is invariantive 31, 32 
Associated forms 148, 157 Ex (v) 
Ausdelmungslehre 279 

Baker 298 
Bezout 318 
Brianchon 280 

Canonical forms 233, 234, 235 
for quintic 231 
for sextic 231, 232 



Capelli 359, 360 

Catalecticant of binary forms of even 
order 232 

of ternary qaartic 313 
Cayley 3, 84, 131, 365 
Chemico-algebraic theory 366, 367, 368 
Ciamberlini 274 
Circle, director 302 

self-conjugate 302 
Class 255 
Clebsch 31, 132, 138, 147, 150, 157, 255, 

265, 271, 274, 326 
Clifford 288 
Cogredient 

transformation defined 36 

sets of variables, forms with 40, 60, 
61 
Combinants 314-318 
Complete systems 85, 109 

relatively 109, 110 

completeness of svstem derived from 
two 111, 113, 114 
Complex variable 208-212 
Concomitant, Mixed 255 
Conic 

Binary quadratic and geometry of 
235-240 

Ternary quadratics and geometry of 
two 288-294 

Polar 251, 256 

Apolar, see Apolarity of ternary forms 
Conjugate lines 300, 302 

points 300 
Contragredience 

defined 36 

Fundamental property of 37 

of point and line coordinates in ternary 
forms 255 
Contravariant 255, 256 
Convolution defined 40, 41 
Coordinates, Line 255, 361 
Copied forms 80, 81, 82 
Covariants of Binary forms 

defined 8 

Symbolical expression of 16, 17, 18 

with several sets of variables 40 
Covariants of ternary forms. Symbolical 

expression of 250 



382 



INDEX 



Covariants 

of degree three 62-71 
of degree four 71-74, 76, 77 
Cross ratio 
defined 186 
cross ratios of roots of a quantic and 

the invariants 189-191 
of the roots of two quadratics 192, 

193 
of the roots of a quartic 204-206 
Cubic, Binary 
Irreducible system for one 89-91, 122 
Irreducible system for quadratic and 

164, 165 
Irreducible system of types of 323-325 
Syzygy for 96 

Transvectants of 96, 97, 194 
Geometry of 193, 194 
Pencil of cubics K/"+\H 194-197 
Geometry on Argand diagram of 211, 

212 
Eange formed by three points and one 

of their Hessian points is equian- 

harmonic 196, 211 
Associated forms for 157 
Pencil of cubics having a given Ja- 

cobian 241, 242 
Cubic apolar to three given cubics 

220, 221 
Rational plane cubic curve represented 

by 221-224 
Twisted cubic curve and binary cubics 

240-242 
Cubic, Ternary 
has three linearly independent apolar 

conies 304 
Canonical form for 312, 313 
Cyclically projective range 196 

Darboux 306 
Degree defined 

for binary forms 9 
for ternary forms 255 
Determinant formed by coefficients of 
7J + 1 binary n-ics is a reducible 
invariant if n is odd 350 
is an irreducible invariant if n = 4 350 
Differential equation whose coefficients 
are polynomials can be reduced to 
a relation between transvectants 
373 
Diophantine equations 102-106 
Discriminant 

of binary forms 191 
of binary quadratic 86 
of binary cubic 194 
of binary quartic 199 
Tangential equation of curve found 
from the 266, 267 
Double points 223, 224 
Duality, Principle of 256, 257, 267 

Elliott 61, 150, 167, 366 



Equianharmonic range formed by cubic 
and one of its Hessian points 196, 211 

Faltung 40 
Forsyth 264, 265 
Franklin 131 
Fundamental theorem 

for binary forms 25-27, 31-35 

for ternary forms 258-264 

Genei-ating function 330 
Gordan 61, 67, 68, 85, 128, 131, 132, 
138, 150, 169, 274, 294, 326, 360, 
367, 368 
Gordan's series 

Existence proved 45, 46 

Coefficients of 53-55 

Inversion of 59, 60 

Deductions from 55-57 
Gordan's proof of Hilbert's lemma 178- 

182 
Gordan's theorem 101, 118-121 
Grace 326, 378 
Grade 

defined 62 

of covariants of degree three 66, 67 

of covariants of degree four 77 

Classification by 119-121 
Grassmann 279 
Groups, Continuous 6, 7, 38 

Finite substitution. Introduction to 
339-344 

Positive symmetric group 345, 346, 
357 

Negative symmetric group 345-351 
Gundelfinger 132, 232 

Harmonic 

quadrilateral 301 
range 191, 192, 203, 207, 272 
Harmonically eoncyclic points 209 
circumscribed conic 300 
inscribed conic 300 
Hesse 306 
Hessian 
defined 3 

is a covariant 3, 16, 17 
identically vanishes when quantic is a 
perfect Jith power and conversely 
193, 235 
points of a cubic 195, 196, 207, 211, 

212 
of binary cubic defines the double 
point on the corresponding cubic 
curve 224 
of ternarv forms 249 
Hilbert 145, 169 et seq., 274 
nomographic ranges 187, 188 

transfoimation 253 
Hyperdeterminants 83, 84, 365 

Identities, Fundamental 

for binary forms 19, 321, 368 



INDEX 



383 



Identities, Fundamental (cont.) 

for ternary forms 257, 258 

in general 349 
Inflexion, Points of 222-224 
Invariants 

defined 7, 8 

of binary forms, Symbolical expres- 
sion for 14, 15, 16 

of ternary forms, Symbolical expres- 
sion for 250, 251 
Inversion 208 
Involution 203, 238 
Irreducible system of forms 85 

solutions of Diophantine equations 
102-105 

Jacobian 

of binary forms 8, 77 

is a covariant 8, 9 

of a Jacobian and another form is in 
general reducible 78 

Product of two 20, 78-80, 140 

vanishing of 193 

of two quadratics is harmonic to both 
192 

of pencil of cubics 241, 242 

of two quartics apolar to a third 
245 

Any combinant of two binary quad- 
ratics is an invariant of their 
Jacobian 315 

of ternary forms 249, 251 
Jordan 62, 126, 321, 375, 377 

Kempe 366 
Klein 374 

Lie 38, 366 

Line coordinates 255 

Linear forms 

Invariants of a binary system of 
33-35, 85, 86 

Invariants of any finite binary system 
and a 160, 161 

Invariants of a ternary system of 275 

Geometry of 279 

with any number of variables 346 

MacMahon 326, 330 
Maximum order 337 
Meyer 245, 317, 368, 374 
Mixed concomitants 255 

Number of independent co variants of 
degree three and given weight 65, 
67 

Operator fi for binary forms : 

Effect on symbolical product 23-25 
Transvectauts defined by 46 
and generalised transvectauts 83 
and hyperdeterminants 84 
Expansion of fi'" { {^v)"P} 173-175 



Operator for ternary forms : 

Effect on symbolical products 259-261 

and transvectauts 295, 296 
Order 9, 255 
Osculating plane 188, 243 

Parametric representation 236, 305 
Pascal 240, 280 
Peano 821, 358, 359 
Pencil 

of rays 188, 189 

of cubics {Kf+ \t) 194-197 

of cubics (Xi/i-f Xa/^ 241, 242 

of quartics 197-208* 
Perpetuants 326-330, 338, 378 
Polar conic 251, 256 
Polar forms 

Definition and symbolical expression 
of 12, 13, 345, 357 

are covariants 39, 40 

Adjacent terms of 44 

Difference between two terms of 42- 
55 

Expansion of one term in 55 
Poncelet's porism 245 
Porism 245, 288 
Projection 253 

Quadratics, Binary 
possess invariants 1, 2 
Complete system for one 86, 121 
Complete system for any number 161, 

162 
Syzygies for any number 162-164 
Covariant Types of 321-323 
Associated forms for one 157 
Geometry on straight line for 191-193 
Geometry on Argand diagram for 

209-211 
and geometry on a conic 235-240 
Complete system for cubic and one 

164, 165 
Complete system for quartic and one 

168 
Any finite system and one 165-168 
Combinants of 315 

Quadratics, Ternary 

Complete system for one 275-277, 298 
Complete system for two 280-286 
Geometry of two 288-294 (see also 
Apolarity) 

Quadratic, Quaternary 
Complete system for 270 
Geometry of 271-273 

Quadrics 272 

Quantics defined 3 

Quartic, Binary 

Complete system for 91-94, 122 
Transvectauts for 97-100, 197 
Syzygy for 98, 200 
Discriminant of 198, 199 
Quadratic factors of sextic covariant 
of 200-203 



384 



INDEX 



Qnartic, Binary {cont.) 
Associated forms for 157 
Geometry and pencil of 197-208 
Cross ratio of roots of 204-207 
defines a pencil of cubics 241, 242 
defines a twisted quartic curve 242- 

245 
Irreducible invariant of five quartics 
350 

Quartic, Ternary 

condition for an apolar conic, 304 
Canonical form for 313 

Quaternary forms 269-273, 349, 362 

Quintic, Binary 
Elementary transvectants of 94, 95 
Complete system for 128-132 
Linear covariants of 133 
Invariants of 133, 136 
Values of transvectants of 135, 137- 

140 
Syzygies for 143, 144 
Typical representation of 145-147 
Associated forms for 148-150 
Canonical form of 229-231 

Eeciprocation 291 
Rectangular hyperbola 303 
Eepresentation, Typical 145-147 
Reye 245 
Richmond 245 
Rosanes 245 

Salmon 365 
Schlesinger 307 
Scott 367 
Self-conjugate circle 302 

triangle 238, 288 
Seminvariant 28-30 
Sextic, Binary 

Complete system for 150-156 

Canonical form of 231, 232 

Associated system for 157 
Straight line, Geometry on 183-189 
Stroh 71, 140, 142, 144, 152, 164, 326 
Stroh's series 

obtained 62-64 

deductions from 65-71 

scheme used for 68 

used 99, 100, 328, 329, 336 
Study 366,368 
Stlife 62 
Stunn 245 

Superosculation, Points of 244 
Sylvester 131,366 



Syzygies 

for Quadratics 162-164 

for Cubic 96, 194 

for Quartic 98, 198 

for Quintic 143, 144 

Stroh's method of obtaining 140-143 

Reducibility of 144 

of second and higher kinds 145 

Finiteness of system of 177, 178 

Tangential equation 301 
Transformations, Linear 

defined 4 

form a group 5, 6 

of binary symbolical pi'oduct 13, 14 

Geometrical interpretation of 184-189, 
208, 251-254 

of ternary symbolical product 247, 248 

Modulus or determinant of 1, 2, 14-17, 
21-23, 248, 249 
Translation principle 

for ternary forms 265-268 

for quaternary forms 271-273 
Transvectants 

defined 46 

Calculation by means of polar s of 
47-49 

Adjacent terms of 49 

Difference between two terms of 49-52 

System obtained by transvection from 
two finite systems 107-109, 111 

Generalized 83 

of ternary forms 296, 297 
Triad 194 
Types, Covariant 

of quadratic 162, 321-323 

of cubic 323-325 

of binary n-ics 335, 336, 379 

Perpetuant 326-330 
Tvpical representation of binary quintic 

145-147 

Ueberschiebung 46 
Unicursal curve 188 

Variables, Different species of 361, 362 
Von Gall 326 

\Webb 186 
Weber 173 
;®eight 15, 17, 28 
Wronski 233, 370-372 



Young 331,356,360 



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